Notes

`sum_(n=0)^oo (2n)! x^(2n)/(n!)` To find radius of convergence of a series `sum` `a_n` , apply the Ratio Test.  `L = lim_(n->oo) |a_(n+1)/a_n|` `L=lim_(n->oo)| ((2(n+1))! x^(2(n+1))/((n+1)!))/((2n)! x^(2n)/(n!))|` `L=lim_(n->oo) | ((2n+2)!)/((2n)!) * (x^(2n+2)/((n+1)!))/(x^(2n)/(n!))|` `L=lim_(n->oo) | ((2n+2)!)/((2n)!) * x^(2n+2)/((n+1)!)*(n!)/x^(2n)|` `L= lim_(n->oo) | ((2n+2)(2n+1)(2n)!)/((2n)!) * x^(2n+2)/((n+1)n!)*(n!)/x^(2n)|` `L=lim_(n->oo) | ((2n+2)(2n+1)x^2)/(n+1)|` `L=lim_(n->oo)|(2(n+1)(2n+1)x^2)/(n+1)|` `L=lim_(n->oo) |(2(2n+1)x^2|` `L=|2x^2|lim_(n->oo) |2n+1|` `L=|2x^2| * oo` `L=oo` Take note that in Ratio Test,  the series diverges when L > 1. So the series diverges except at x=0. Since the series converges at x=0 only, therefore, the radius of convergence is R=0 .

`int(x^2-1)/(x^3+x)dx` `(x^2-1)/(x^3+x)=(x^2-1)/(x(x^2+1))` Now let's create partial fraction template, `(x^2-1)/(x(x^2+1))=A/x+(Bx+C)/(x^2+1)` Multiply equation by the denominator, `(x^2-1)=A(x^2+1)+(Bx+C)x` `(x^2-1)=Ax^2+A+Bx^2+Cx` `x^2-1=(A+B)x^2+Cx+A` Comparing the coefficients of the like terms, `A+B=1`  ----------------(1) `C=0` `A=-1` Plug the value of A in equation 1, `-1+B=1` `B=2` Plug in the values of A,B and C in the partial fraction template, `(x^2-1)/(x(x^2+1))=-1/x+(2x)/(x^2+1)` `int(x^2-1)/(x^3+x)dx=int(-1/x+(2x)/(x^2+1))dx` Apply the sum rule, `=int-1/xdx+int(2x)/(x^2+1)dx` Take the constant out, `=-1int1/xdx+2intx/(x^2+1)dx` Now evaluate both the integrals separately, `int1/xdx=ln|x|` Now let's evaluate second integral, `intx/(x^2+1)dx` Apply integral substitution: `u=x^2+1` `du=2xdx` `=int1/u(du)/2` `=1/2int1/udu` `=1/2ln|u|` Substitute back `u=x^2+1` `=1/2ln|x^2+1|` `int(x^2-1)/(x^3+x)dx=-ln|x|+2(1/2ln|x^2+1|)` Simplify and add a constant C to the solution, ...

If you were writing an essay about why Lyddie should not sign the petition, you would want to include information on how Lyddie felt about the petition.  This shows her state of mind, and the information about the consequences of signing it.  Support your arguments with quotes from the book. Lyddie is afraid to sign the petition because she values her job at the factory. She needs the money to pay off her family’s debts.  She hopes that one day she will be able to get the farm back and get all of her family back together. Lyddie got her information about the petition from the other girls in her boarding house.  The petition is for a shorter work week for the girls. "Time is more precious than money, Lyddie girl. If only I had two more free hours of an evening-what I couldn't do."  "Should you sign the petition, Betsy, they'll dismiss you. I know they will." Amelia folded the handkerchief and handed it back to Lyddie with a nod. (Ch. 12)  Lyddie also agrees w...

The end of Ernest Hemingway's "The Snows of Kilimanjaro" is ambiguous to be sure, and it can be difficult to understand whether or not Harry's trip in the airplane was real or imagined. At first, we read a section in which Harry is carried onto a plane and taken away, ostensibly for medical attention. However, rather than heading toward a city, Harry's plane instead flies toward Mount Kilimanjaro. Directly after this section, we read of Harry's wife, Helen, discovering Harry's dead body. While this sequence of events is confusing, it's important to understand that Hemingway is suggesting Harry did not die in the plane, or even travel on the plane at all; rather, he died in the tent, and the trip in the plane was either an illusion or a feverish dream.  It's possible to interpret this ending in a variety of ways. On the one hand, it's possible to see it as a pessimistic conclusion. Rather than escaping and getting medical treatment, Harry dies a...

1. "Like, hey, who's the midget? And, there goes Mad Max; and, excuse me while I barf; and, look what escaped from the freak show; and, oh my gawd that's disgusting ." This is what runs through Max's mind on the first day of school, in Chapter 12, when he and his new best friend Kevin ("Freak") are roaming the halls together, Freak riding on Max's shoulders. He's recounting all the nasty things that the other kids are saying (or perhaps only thinking) when they see the oversized Max and the shockingly small Kevin walking together in that way. The conflict is on display in the quote above, revealing how other people keep on judging both friends based on their appearance: Max, based on his hulking size and his resemblance to his father the felon, and Kevin, based on his small size and his leg brace. Their peers reject Max and Kevin as "disgusting" and a "freak show," worthy of "barf" rather than acceptance. This is a ...

Act 1, scene 2 of Shakespeare's  The Merchant of Venice  is when Portia and Nerissa discuss the different suitors who have come to court to play the game of chance for Portia's hand. Portia has not been impressed with anyone so far. Nerissa names them off in this scene, and Portia gives witty and informative reasons as to why she doesn't like each one. She is witty because she can joke about each suitor's character and personality to make Nerissa laugh, but she also shows her intelligence because she can peg each one exactly. In fact, Portia is so good at categorizing each man that by today's standards, she might be considered prejudiced because she pokes fun at their stereotypical vices. For example, when Portia criticizes the man from England, she makes fun of the fact that he doesn't know any other language than his own. The English would conquer a nation and force everyone to speak English, so Portia plays on that with her comments. Of the Frenchman, she say...

Ah, this question has beset historians for centuries! Theories about what brought the Fall of the Roman Empire include excessive military spending, lead poisoning, and just about everything in between. The present consensus, based upon historical and archaeological evidence, is that there was no singular cause of the Fall, but a number of factors which served to compound one another.  Perhaps the biggest reason contributing to the Fall of Rome was how widespread the Empire grew to be. At its fullest extent, the Roman Empire included the coast of North Africa, all of Southern and Western Europe, and even territory in the Arabian Peninsula. In all of these territories, soldiers and officials were sent from the central Roman territory to act as overseers and enforcers of the Roman law. Over such a wide expense of territory, it was incredibly difficult to govern such various locations with differing troubles like crop failure, crime, or invaders from outside the Empire. Imagine trying to h...

The first chapter of History of Plymouth Plantation is an account of how the Reformation proceeded in England. Broadly, Bradford describes the struggle between Catholics, who he calls the "gross darkness of papacy," and what he calls "orthodox and true Christians" that took place across Europe. Focusing on events in England, he claims that Satan himself was at work during the reign of Catholic Queen Mary, who persecuted Protestants. Under Elizabeth, some reforms were instituted, but the Church retained some of the ceremonies (deemed impious by Calvinist-inspired Englishmen) in order, Bradford says, to "win the weak and ignorant." A group, known as Puritans for their desire to "purify" the Church emerged under James I, and these people were persecuted by the established Church of England. So they, as many dissenters had done under Queen Mary, decided in 1607 to leave England and seek asylum in the Netherlands. The crucial underlying theme establis...

In Raising the Bar and Closing the Gap: Whatever it Takes,  the authors argue that professional learning communities (PLCs) can be instrumental in a school's attempts to reach all of its students.  Evidence of this lies in the way leadership teams work with teachers and how time is managed in the course of a day. DuFour and his team display different examples of evidence of PLC effectiveness in the desire to raise student achievement.  One such instance is seen in chapter three in the discussion of the "educational lottery."  When attempts to reach all of a school's students are not institutionally embedded,   school-wide success decreases.  The authors argue that a PLC is instrumental to student achievement. When initiatives are not widely embraced, then students are subject to the whims of individual teachers.  Some teachers might be practicing techniques and strategies that work. Others might not.  This makes students subject to an "educational lottery" a...

The series given in the problem is `sum_(n=1)^oo 2/10^n` . This is a convergent series. It is a geometric series with first term `2/10` and common ratio `1/10` . The sum of `sum_(n=1)^oo 2/10^n ` = `(2/10)/(1 - 1/10)` = `(2/10)/(9/10)` = `2/9` The required sum `sum_(n=1)^oo 2/10^n = 2/9`

Ponyboy talks about his father on pages six and seven. The Outsiders has been published in various editions, so there is no guarantee that my edition and your edition are the same. I can guarantee that Ponyboy talks about his father in chapter one. There is not a lot of information about Ponyboy's father, either. Ponyboy's information about his father is about a paragraph in length, and the information that Ponyboy gives is used to further highlight Darry's characteristics. Ponyboy mentions how much Darry looks like their father. Ponyboy apparently does not feel he has a reason to write much about his parents, who died eight months earlier and so do not appear in the story. Darry has been taking care of the family ever since their parents' deaths, and he definitely struggles with the responsibility.

Though the number varies due to factors like health status, age, and stress on the hair follicle, physicians generally agree that between fifty and one hundred hairs may be lost from the scalp each day. There are around one hundred thousand hairs on a person's head, and these hairs are constantly going through a cycle of growth and "rest." It is during the resting phase that hairs may fall out. As people age, hair follicles may slow down or stop producing new hairs to replace those that have fallen out. Certain health conditions, like alopecia, can also affect a person's ability to grow hair.  Excessive hair loss (more than 125 hairs per day) may be a sign of a health condition or aging. To avoid rapid or excessive hair loss, be sure to develop healthy stress management behaviors, drink lots of water for healthy skin and blood, and avoid wearing hairstyles that tug on the scalp.

Pathos, ethos and logos are the three pillars of persuasion. It is difficult to be persuasive in writing or speech without using all three. Pathos is the appeal to emotions, ethos is the appeal to credibility or authority (do you believe the writer's or speaker's claims?) and logos is the appeal to logic, usually in the form of facts or statistics. Although  Animal Farm  is a fable about animals, it uses all three forms of persuasion. A chief example of pathos in this novel is Boxer, the strong, dedicated horse who is a true believer in the animal revolution and works harder than everyone else his whole life in honest, whole-hearted support of the dream. It is difficult not to be moved emotionally when he is betrayed in old age: rather than the retirement he has been promised, he is sold to the glue factory. Orwell achieves ethos by making his animal characters behave like real, believable human beings and by basing his book on events in Stalinist Russia, such as show tria...

Montag's wife and her friends represent the hedonist society in which they live. The women come over to visit Mildred and to watch a show on the three television walls she has in her parlor. Disgusted with them and their behavior, Montag turns off the parlor walls and leaves the women sitting there. He decides to ask them questions that might stir their souls to reflection on the state of their lives.  First, Montag asks the guests when they think the war will start and then mentions their husbands. Mrs. Phelps says that her husband was deployed by the army, but he should be back in a couple of days. She's not worried. She also says that since they both are each other's third spouse, they vowed not to cry or hug when he left. Then, going along with the disposable attitude of the society, Mr. Phelps told his wife to forget him quickly and remarry if he dies.  Next, when the discussion turns to children, Mrs. Phelps says that "children are ruinous" (96) and that Mrs...

Despite the fact that Bud believes him to be so for almost the entire duration of the book Bud, Not Buddy , Herman Calloway is not Bud's father. Rather, he is his grandfather.  Bud is under the impression that Calloway is his father because his deceased mother, Angela, left behind several posters showing Calloway with his band, the Dusky Devastators of the Depression. Bud embarks on a wild journey to seek out Calloway and to inform him of their biological connection. However, Calloway is extremely resistant to the idea that Bud is his son, and when Bud's mother's name is eventually revealed to Calloway, he has a major emotional breakdown. Bud learns that his mother was the daughter of Calloway and that she had run away from home because of Calloway's strict parenting style. Calloway believes that Angela ran off with one of the men who used to be in his band; he did not even know that his daughter had passed away. Regardless, Calloway decides to give Bud a home and to of...

Magnetic reversal or geomagnetic reversal is the process in which a planet's magnetic poles reverse their polarity. That is, the magnetic reversal on Earth will mean that our magnetic North pole will become the magnetic south pole and magnetic south pole will become the magnetic north pole. It also means that our compasses will be off by 180 degrees and will point in exactly the opposite direction (north instead of south and vice-versa).  It is a fact that magnetic reversal is a common process and has taken place several times during the history of Earth. The last known instance was about 800,000 years ago.  Before we think about the pros and cons of this process, one has to understand that the process takes a long time and the field does not change overnight. This means that living beings will have sufficient time to adapt to this change (thousands of years). Thus, any notion that magnetic reversal may be dangerous to birds (such as pigeons) who use Earth's magnetism for direc...

Ultimately, I believe young Goodman Brown's willingness to step forward in this passage is internally driven.  It is true that he is struck by the fact that people he'd believed all his life to be pious and righteous individuals sit together, side by side, with known criminals and sinners.  However, "he was himself the chief horror of the scene" because "The fiend in his own shape is less hideous, than when he rages in the breast of man."  In other words, the devil himself is far less frightening and awful than when he prompts men who might have chosen to be good to do evil.  This propensity toward sinfulness already existed in Brown's nature.  Remember, he chooses to enter the forest and meet with the Devil, thinking he will do so on just this one more night and then he'll be good and "'cling to [his wife, Faith's] skirts and follow her to heaven.'"  He is only too ready to lay down his faith, believing that it will be there fo...

We have a Reference tutor online right now to help you! To chat with a tutor, please set up a tutoring profile by creating an account and setting up a payment method. If you initiate a chat, please note you will be charged $0.50 a minute for tutoring time.

This story was first published in 1890, some 25 years after the end of the American Civil War.  Though the story takes place during the time of the war, and though it presents a Southern plantation owner and secessionist who is wholly devoted to the Southern cause, it does not paint him as pure villain.  In fact, Bierce gives the reader one major way in which to relate to Peyton Farquhar: all he wants is to escape his death sentence and return home to be with his wife and children.  He wants this so much that he actually imagines, in the time it takes him to drop from the bridge into the noose, that he has almost achieved it.  Farquhar isn't presented as particularly villainous at all (any details that would draw attention to his slave-ownership, for example, are left out of the story), but rather, he seems like a pretty normal guy with pretty normal hopes.  Therefore, the story seems to focus, not on the villainy of the South or the horrors of slavery as earlier literature was mor...

Recall that indefinite integral follows `int f(x) dx = F(x) +C` where: `f(x)` as the integrand function `F(x)` as the antiderivative of `f(x)` `C` as the constant of integration. The given integral problem: `int x arcsin(x)dx `  resembles a formula from integration table. We follow the integral formula for inverse sine function as: `int x arcsin(ax) dx = (x^2arcsin(ax))/2-arcsin(ax)/(4a^2)+(xsqrt(1-a^2x^2))/(4a)+C` Applying the integral formula inverse sine function with `a=1` , we get: `int x arcsin(x) dx = (x^2arcsin(1*x))/2-arcsin(1*x)/(4*1^2)+(xsqrt(1-1^2x^2))/(4*1)+C`                                 `= (x^2arcsin(x))/2-arcsin(x)/4+(xsqrt(1-x^2))/4+C`                                          or  `((2x^2-1)arcsin(x)+xsqrt(1-x^2))/4+C`

Although Patrick Henry delivered his “Give Me Liberty or Give Me Death” speech before the Declaration of Independence had been written and signed, the issue that he addresses in this speech is, essentially, the issue of American independence.  He is saying that the colonists can fight (and risk dying) for freedom or can choose to live in slavery.  Henry delivered this speech in March of 1775.  The colonies had, of course, not actually declared independence.  The battles at Lexington and Concord had not yet occurred.  The Virginia Convention was not discussing whether to declare independence.  However, this was the issue that Henry was talking about. The actual subject of debate was whether to organize militias in every county of Virginia to fight, if needed, against the government.  Henry had proposed this, but some delegates thought that it was a step too far or that it was too provocative.  Henry did not agree.  He argued that the colonists had tried everything they could to resolve ...

One leader who embraces the Transcendental concept of Individualism as promoted in Ralph Waldo Emerson's essay "Self-Reliance" is Donald J. Trump. Another leader in the business world is Sir Richard Branson, who has also fought against imitation like Emerson did. There is no question that Trump is his own person and believes in Emerson's motto, "Trust thyself." For instance, during one interview with Bill O'Reilly, O'Reilly cautioned Trump regarding broaching certain topics in his campaign rallies and doing certain actions. Trump's response was to say, "Bill, I'm a winner." He trusted himself and his ability to win in the Republican debates, and succeeded in becoming the party's nominee for President of the United States. Another leader in the recording industry, airlines, and spacecraft, who is very self-reliant and does not keep pace with his companions. . . because he hears a different drummer (Henry David Thoreau) is Sir Ric...

`intcos^3(x)sin^4(x)dx` Rewrite the integral as , `intcos^3(x)sin^4(x)dx=intcos^2(x)cos(x)sin^4(x)dx` Now use the trigonometric identity:`cos^2(x)=1-sin^2(x)` `=int(1-sin^2(x))cos(x)sin^4(x)dx` Apply integral substitution: `u=sin(x)` `du=cos(x)dx` `=int(1-u^2)u^4du` `=int(u^4-u^6)du` Apply the sum and power rules, `=intu^4du-intu^6du` `=(u^(4+1)/(4+1))-(u^(6+1)/(6+1))` `=u^5/5-u^7/7` Substitute back u=sin(x)  and add a constant C to the solution, `=(sin^5(x))/5-(sin^7(x))/7+C`

Large numbers can be expressed in standard form, also known as scientific form, by using positive integral exponents.A number is in standard form when it is written as `a*10^n` , where a is a decimal number greater than or equal to one and less than 10, `(1<=a<=10)` and n is a positive integer. Similarly, small numbers can be written in standard form using negative integral exponents, written as `a*10^(-n)` Now let's express the stated numbers in scientific form: 1. `70,000,000,000=7*10^10` 2. `0.000000048=4.8*10^(-8)` 3. `67,890,000=6.789*10^7` 4. `70,500=7.05*10^4` 5. `450,900,800=4.509008*10^8` 6. `0.009045=9.045*10^(-3)` 7. `0.023=2.3*10^(-2)`

Curtis plants tomatoes in the lot in order to impress Lateesha, the girl he wants to marry. Accordingly, Lateesha had stopped seeing Curtis five years ago when she discovered other girls in his life. For his part, Curtis admits that he has now sobered up and is serious about marrying Lateesha. The only problem is that she won't give him the light of day. So, knowing how much Lateesha enjoys eating tomatoes, Curtis plants six Beefsteak tomato seedlings in the garden lot right across from her third-floor apartment. He weeds, fertilizes, and picks the bugs off the tomato plants diligently. In due time, the tomatoes grow abundantly. Curtis also stops going about shirtless; his goal is to show Lateesha that he is a man of refinement, despite the fact that he has muscles. To Curtis, growing tomatoes is the only way he thinks he can show Lateesha that he's serious about being a good candidate for a husband. In due time, Curtis puts up a sign in front of the tomatoes that says "La...

The quick answer is that you need conditions where (1) females mate with multiple males, (2) different variants of the trait must result in different degrees of reproductive success, and (3) these variants must be heritable. Let's unpack what all that means by going into more detail. First I'll define sperm competition, and then give examples of the sorts of traits that might evolve by sperm competition.  Sperm competition is competition "between males to prevent each other from fertilizing eggs" (Krebs and Davies, below). It is a type of male-male competition, which, in turn, is a type of intra-sexual selection, i.e., sexual selection that consists of members of one sex competing for access to the other sex. When we hear the words "sperm competition," we might think first of sperm battling each other in the reproductive tract of a female who has mated with multiple males in succession. We envision traits specific to the sperm themselves. While sperm competi...

This statement follows a difficult argument between Macbeth and Lady Macbeth. Lady Macbeth becomes very angry with her husband when Macbeth says he does not want to move forward with their plan to kill Duncan (1.7.34). Lady Macbeth proceeds to try all manner of insults and manipulation to coerce Macbeth to recommit to their plan, including wounding his ego by insulting his masculinity and implying he is disloyal to her. She calls Macbeth a "coward" and claims he lacks courage. Lady Macbeth also says she would kill her own baby if she had promised him that she would do so. Once Lady Macbeth convinces Macbeth that no one will doubt their feigned grief when Duncan is discovered dead and that they can actually get away with this murder, Macbeth says, "I am settled and bend up / Each corporal agent to this terrible feat" (1.7.92-93). In other words, Lady Macbeth has convinced Macbeth to be in favor of their plan. After much indecision, Macbeth is no longer on the fence. ...

Energy is transferred through an ecosystem as organisms eat and are in turn eaten by other organisms. The original energy source in most ecosystems is the energy of the sun, which is captured by primary consumers, or autotrophs. These are green plants and algae capable of carrying out photosynthesis. During this chemical reaction, light energy is converted to chemical energy in the form of glucose. Other organisms called primary consumers or herbivores, will eat the producers. Secondary consumers, which are carnivores, eat the primary consumers and obtain energy. Therefore, they indirectly get energy from the sun when they consume their prey. As the trophic levels increase from producer, to primary consumer, to secondary consumer, and so on, the amount of energy decreases at each level. If 100% of available energy is at the producer level, the primary consumers will get 10% of that energy when they consume that producer, and secondary consumers will get 1%. This loss of energy is due t...

Judge Taylor in Lee's  To Kill A Mockingbird  takes his job seriously. He is the only judge in a racist community when a very controversial case crosses his desk. Rather than play to the white majority's wishes to convict Tom Robinson without much of trial, Judge Taylor appoints the best lawyer in the county to defend a black man. He knows Atticus Finch will provide the best possible defense for Tom Robinson even though the community might hate him for it. He does this because he believes everyone deserves a fair trial. The book doesn't come out and say these things explicitly, though. Most of this can be inferred by what Atticus says to his brother about his appointment as follows: "Before I'm through, I intend to jar the jury a bit—I think we'll have a reasonable chance on appeal, though. I really can't tell at this state, Jack. You know, I'd hoped to get through life without a case of this kind, but John Taylor pointed at me and said, 'You're...

"In a Grove" is perhaps the most famous of Akutagawa Ryunosuke's works. It is the basis for Kurosawa Akira's movie Rashomon and helped cement the widely used plot line of telling varying accounts of solving a crime, a plot that is seen on both television and in film. However, using this method of storytelling also means that it can be difficult to know which details are true. It is obvious to blame Tajomaru for the murder of Takehiro, as he is the criminal in this story. He admits to the crime. However, there are several inconsistencies within his confession that do not seem to reflect Takehito truthfully as a person. As is sworn by Masago's mother, Takehito's mother-in-law, he was a person with a gentle nature, contrary to the greedy person Tajomaru describes. This is where the reader must begin to draft their own version of events, as it seems the testifying characters are lying. Afterwards, Masago herself takes responsibility for the crime. Her description...

Jimmy Valentine is a hardened criminal who immediately returns to safecracking when he is released from prison; however, after he unexpectedly falls in love, Jimmy changes his behavior and attitudes and becomes an honest businessman. But, later he must return to his former self, although it is only briefly. As the story begins, Jimmy Valentine is handed his pardon by the warden, who urges him to "[B]race up, and make a man of yourself...Stop cracking safes, and live straight." Jimmy rejects this advice with a bold-faced lie: "Me?....Why, I never cracked a safe in my life." Then, as soon as he walks out of the prison, Jimmy takes a train to a small town near the state line. There he makes contact with an old friend, who has helped to procure Jimmy's pardon from the governor. After a brief conversation, Jimmy asks him for the key to his old room. Opening his room, where the detective's collar-button yet lies on the floor from the day on which he was arrested, ...

Roe v. Wade was an important Supreme Court case decided in 1973 that allowed women across the United States the right to an abortion during the first two trimesters of pregnancy. At that time, many states restricted abortion rights. The court based its decision on the right of privacy, given under the Due Process Clause of the 14th Amendment of the Constitution. The justices felt that a woman's right to privacy was more compelling than the right of states to regulate that right during the first and second trimester of pregnancy. Around the third trimester of pregnancy, which the court determined to be the point of viability (or when the fetus could survive on its own), the court determined that the states can regulate or restrict a woman's rights to an abortion, with exceptions to protect the life or health of the mother.  This case has been contentious since the Supreme Court passed down its decision in 1973. Supporters believe a woman's right to choose whether or not to ...

Limit comparison test is applicable when `suma_n` and `sumb_n` are series with positive terms. If `lim_(n->oo)a_n/b_n=L` where L is a finite number and `L>0` , then either both series converge or both diverge. Given series is `sum_(n=1)^oosin(1/n)` Let the comparison series be `sum_(n=1)^oo(1/n)` The comparison series `sum_(n=1)^oo1/n` is a p-series of the form `sum_(n=1)^oo1/n^p` with p=1. p-series test states that `sum_(n=1)^oo1/n^p` is convergent if `p>1` and divergent if `0<p<=1` So ,the comparison series is a divergent series. Now let's use the limit comparison test with:`a_n=sin(1/n)` and `b_n=1/n` `a_n/b_n=sin(1/n)/(1/n)` `lim_(n->oo)a_n/b_n=lim_(n->oo)sin(1/n)/(1/n)` Let's apply L'Hopital's rule to evaluate the limit. Test L'Hopital's condition: `0/0` `=lim_(n->oo)(d/(dn)(sin(1/n)))/(d/(dn)(1/n))` `=lim_(n->oo)(cos(1/n)(-1/n^2))/(-1/n^2)` `=lim_(n->oo)cos(1/n)` `lim_(n->oo)1/n=0` `lim_(u->0)cos(u)=1` `=1>0` Since th...

I find that I procrastinate most about the tasks I dislike doing, and I think it's likely that is true of most people. Therefore, one of my most successful methods of dealing with procrastination is to first do the task that I least want to do. This has two advantages.  First, the task I have procrastinated about is done, and second, for the rest of my day, I do not have that task hanging over my head, draining my energy and making me feel guilty about what is undone.  For example, when I practiced law, I never minded doing research and writing briefs, but I hated to write boilerplate complaints and answers.  So I always got those out of the way.  Now that I teach, I find that I dread having to grade handwritten work, as opposed to work that is typed.  I do the handwritten quizzes first and then reward myself with the typed quizzes.  When you have schoolwork to do, first do the work you like the least. You will be amazed at how much more productive you are. 

In Book V, Athena attempts to influence Zeus on Odysseus's behalf because she feels sympathetic toward Odysseus and his family.  Odysseus has been kept for seven years by the nymph, Calypso, and though he badly wants to return home, she will not let him leave.  Athena argues that Odysseus needs a great deal of help because he no longer has a "crew to ply the oars / and send him scudding over the sea's broad back."  Alliteration is the repetition of an initial consonant sound, and the words send , scudding , and sea all begin with the "s" sound; therefore, the line that contains these words contains an example of alliteration.  This particular repetition seems appropriate, given the subject matter of the line -- water -- because the "s" sound can be used to replicate the sound of moving water: many words that describe wet things begin with "s" -- slurp, spurt, slippery, slushy, and so forth (these are known as phonetic intensives ).  The...

Freezing refers to the phase change from liquid to solid phase. In this case, we are discussing the freezing of water. Water freezes at 0 degrees Celsius. Since no other information is given, let us assume that the water is already at 0 degrees Celsius and is going to freeze. The amount of heat released when the water freezes is also known as the latent heat of fusion and is equal to 80 calories per gram of water or, 334 Joules per gram of water. Here we have 100 grams of water. Thus, the amount of heat released on freezing = mass of water x latent heat of fusion. = 100 g x 334 J/g = 33,400 J = 33.4 kJ   Thus, 33.4 kilojoules of energy is released when 100 g of water freezes.  Hope this helps. 

It looks like you have to put together a small portfolio of diversified stocks. That's a fun assignment! First, my advice is based on what I've taught using the Stock Market Game and the Gen I-Revolution online finance games. It looks like the game portal your teacher is using is similar. I typically recommend diversifying with a combination of mutual funds, fixed-income securities like United States Treasury Bills, mid-cap stocks, and large-cap stocks. Non-cyclical stocks are also a good bet. These are basically stocks that aren't overly affected by market fluctuations. They do well whether the economy is buoyant or tanking (within limits). Examples of non-cyclical stocks are food and utilities stocks from companies like General Mills, Proctor and Gamble, Coca Cola, and Duke Energy . Take a look at top performing utilities stocks here. Compare the list to the top electric utilities stocks of 2016. Whatever you decide to invest in, remember to check the latest news and ...

Arthur and Kate Keller were determined to find help for their daughter, Helen.  Helen became deaf and blind when she was one year old.  In the 1880s, children with such severe disabilities were often sent away to live in asylums or at boarding schools.  They were sometimes not expected to learn or live normal lives.  Helen's parents did not feel this way. As Helen grew older and tried to communicate, Mr. and Mrs. Keller "were deeply grieved and perplexed."  They wanted to help their daughter, but they did not know how.  They began to research options.  They did not live anywhere near a "school for the blind or the deaf."  Helen's mother read "American Notes" by Charles Dickens.  She read about "his account of Laura Bridgman, and remembered vaguely that she was deaf and blind, yet had been educated."  Her father discovered an "eminent oculist in Baltimore."  They traveled there "to see if anything could be done for [Helen...

Just as the fog prevents people from seeing clearly in the literal sense, the fog seems to be presented as symbolic of Scrooge's inability to see in a figurative sense: he cannot see what his real priorities ought to be or that the choices he has made in his life have actually caused him to be alone.  As he walks home on Christmas Eve, the fog is especially thick, just as Scrooge's selfishness and lack of compassion are at their height.  When he wakes up on Christmas morning, one of the first things he notices is that the fog has lifted, and it is a bright and clear day.  Scrooge's mental fog, if you will, has also lifted, and he now understands how he should view his fellows, that he should live to help them in every way that he can, and that he should reach out to his nephew, Fred, his only living family. 

First, Dickens makes the Ghost of Christmas Past part of an effective story by making him a vivid character. For one thing, the Ghost's appearance is quite distinctive and memorable. He wears a white tunic decorated with flowers, holds a sprig of holly, and a jet of light emanates from his head. Further, while he is the size of a child and unwrinkled, he has long white hair. Most curious of all, he fluctuates: sometimes he is whole and sometimes all Scrooge can see is an arm or a leg. The Ghost also has a distinctive personality. He is authoritative and won't let Scrooge, who is unwilling to travel with him, off the hook. Second, he interacts with Scrooge in ways that encourage Scrooge to question his own assumptions. For example, when the Ghost sees how much Scrooge longs for Christmases past, he mocks him using Scrooge' own speech patterns: A small matter,” said the Ghost, “to make these silly folks so full of gratitude. Although that's exactly what Scrooge might have...

Francis Macomber is obviously afraid of the lion. He doesn't want to get out of the safety of the steel body of the car and expose himself to being charged. But he makes one of his many mistakes in asking Wilson, "Why not shoot him from where I am?" Macomber thus shows his fear. Wilson simply tells him, "You don't shoot them from cars." It would be bad sportsmanship. The hunter should at least give the lion some chance rather than just killing it with a high-powered rifle from a position of safety. Hemingway describes Macomber's fear as he has to get out of the car and move closer to the lion. Macomber had not thought how the lion felt as he got out of the car. He only knew his hands were shaking and as he walked away from the car it was almost impossible for him to make his legs move. Hemingway had plenty of experience with big-game hunting in Africa. He wrote a whole book about it titled Green Hills of Africa  (1935). Francis Macomber illustrates one o...

We are asked to graph the function `y=(8)/(x^2-x-6) ` : Factoring the denominator yields: `y=8/((x-3)(x+2)) ` There are vertical asymptotes at x=-2 and x=3. Since the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is y=0. The y-intercept is -4/3. The graph:

In Chapter 8 of The Chrysalids , after David's Aunt Harriet is found dead, clearly having killed herself after having produced a third child that had some "mutation," David begins to worry about himself, coming to an understanding that he is a deviant, too.  He prayed, saying, ....please, please, God, let me be like other people. I don't want to be different. Won't you make it so that when I wake up in the morning I'll be just like everyone else.... David talks with his Uncle Axel after five nights of praying to no avail, and he shares with his uncle what has happened to Aunt Harriet and the baby, which had completely disappeared.  He understands that the baby was taken because it was different, and David, of course, is different as well.  Uncle Axel assures him that he will never reveal his secret or the others', but that David must be very careful to never let anyone else know about his mutation, an ability to communicate with others through thoughts alo...

For years, researchers have been examining the link between violence in the media and its effects on individuals.  Now experts are looking at social media and its possible relationship to violence.  Social media outlets such as Twitter, Facebook, YouTube and SnapChat are being used by gangs to recruit members, communicate with their organizations and participate in verbal feuds with rival gangs.  Social media is also being used to document and communicate acts of violence.  Individuals now regularly record and post videos of fights and other crimes.  Bragging rights go to the video that gets the most likes or views.  Past studies have supported the theory that exposure to violence desensitizes an individual and makes these acts more acceptable.  With social media being another avenue for people to view violence, promotion of violent acts has become even more commonplace in our society.  As an educator, I personally see how social media creates conflict on a daily basis.  Posting on soc...

Mr. Wright was found dead in his bed with a rope around his neck. As the story unfolds, the irony of the way he died becomes clearer. Three types of irony related to the cause of death are apparent. Situational irony refers to events that are the opposite of what one might expect. Dramatic irony occurs when the reader knows something that a character doesn't know. Verbal irony refers to the use of words to mean the opposite of their literal meaning.  First, that Mr. Wright would be strangled in his bed with a rope is ironic--unexpected--for several reasons. There was a gun in the home, and anyone who wanted to murder the man could have done so more quickly and easily with the gun. Second, that he didn't wake up while the rope was being put around his neck is unusual. Third, the method of death was like a hanging, a way to administer justice to a wrong-doer, yet the man was murdered, so the one who was administering the "justice" would be the wrong-doer--making for an ...

There was a shift in the Civil Rights Movement in the 1960s. While the movement had generally been one focused on nonviolent protests, there was more activism in the 1960s. Some African-Americans believed that change would come quicker if more confrontational activities occurred. They believed that non-violent protests weren’t bringing about change quickly enough. They believed that attitudes really hadn’t shifted a great deal, especially in the South. Groups like the Black Panthers and the Nation of Islam advocated stronger actions to try to deal with the issues that were affecting the African-American community. In some instances, this involved confrontational actions. They believed this would bring about change faster than relying on purely non-violent methods. Bobby Seale and Huey Percy Newton founded the Black Panthers. While they weren’t afraid to use violence, they wanted to make life better for the African-American people. They developed a ten-point plan to bring about equality...

We really know very little about Fortinbras since he doesn't appear in person until near the end of the play. From what we learn about him through his actions and the reports of others, Fortinbras seems like a rash, impulsive, ambitious, inexperienced young man who is out to make a name for himself. If he became king of Denmark, as seems very likely to happen, he would probably always be involving the nation in military adventures, much like a young Napoleon Bonaparte. Fortinbras seems very much like young Octavius in Shakespeare's play Julius Caesar . Octavius is also young and inexperienced. He has nothing to recommend him as one of the rulers of Rome except for the fact that Julius Caesar appointed him his heir. Fortinbras is in a similar position. He is the nephew of the reigning king of Norway and heir apparent to that throne. Both Octavius and Fortinbras are young, ambitious, untested, impulsive, and eager to perform great deeds and make names for themselves. By contrast,...

Human understanding of atomic theory has developed vastly, thanks to the contributions of several scientists whose work spanned the early 1800s through the 1920s. The work of Dalton, Thomson, Rutherford, Bohr, and Schrödinger occurred over a short period, and their discoveries advanced atomic theory rapidly. Below is a brief summary of each of their contributions to atomic theory. John Dalton John Dalton was a scientist in the 1800s. Simply, he was the first to describe atoms and their properties. His contribution to atomic theory consisted of five main tenets (1805):  All matter is composed of small particles called atoms. Atoms are the smallest unit of matter; Matter cannot be broken down into units smaller than atoms; Atoms are indestructible. There are different kinds of atoms; atoms of the same kind of element are identical to each other. Atoms of the same element are identical in mass and size and have the same properties. In chemical reactions, atoms are rearranged, combined, or...

We are asked to show that the following is an identity: `sin2A+tan2A=(sin2A*tan2A)/(tanA) ` The correct procedure is to start with one side, and through allowable algebraic manipulations end with the other side. Note that it is impermissible to begin with the "equality" and then use manipulations, e.g. subtract sin2A from both sides, as this implies that the sides were known to be equal. Here we will start with the right side (RHS) and show that it is equivalent to the left side (LHS). `"RHS"=(sin2A*tan2A)/(tanA)=(2sinAcosA*tan2A)/(sinA/cosA) ` `=(2sinAcosA*tan2A)/(sinA/cosA)*(cosA/sinA)/(cosA/sinA) ` `=2cos^2A*tan2A ` `=(2cos^2A-1+1)tan2A ` `=(cos2A+1)tan2A ` `=cos2Atan2A+tan2A ` `=cos2A*(sin2A)/(cos2A)+tan2A ` `=sin2A+tan2A="LHS" ` as required.

A market economy is one in which the prices of goods, investments, and services are unimpeded by government regulation of any sort with the market and laws of supply and demand setting prices. The first thing to note is that a pure market economy has never existed. All governments engage in various practices such as taxation, infrastructure building, subsidies, and other activities which affect pricing. For example, if a government pays for roads via taxation but not for railroad tracks, the government is in effect creating an artificially low price for road vs. rail transport. The good point of a market economy is that it should, in theory, assuming a perfectly competitive market, allow for prices to rise and fall with consumer demand.  Unfortunately, the assumption of a perfect market ignores such issues as monopoly or collusion, i.e. businesses setting prices artificially low to drive competitors out of business and then forming monopolies or oligopolies which keep prices artificial...

We are asked to graph the following function: `y=log_(1/3)(x)` The domain is x>0 and the range is all real numbers. The graph of the function is decreasing and concave up on its domain. Some points on the graph: (1/27,3),(1/9,2),(1/3,1),(1,0),(3,-1),(9,-2) The graph:

While it is somewhat popular to believe that the Thirteen Colonies were economically and politically unified, nothing could be further from the truth. Throughout the colonies, differing political and economic goals often led to conflict.  Even within each individual colony, a strict social stratification was taking place.  An elite class was emerging that was quickly becoming politically powerful and economically advantaged.  It is the emergence of this aristocratic class that led to several rebellions throughout colonial history.  The most famous of the rebellions was led by Nathaniel Bacon in Virginia in 1676.  Disheartened by the lack of opportunity afforded to middling planters, and angered by the governors lack of interest in defending the frontier against Indian attacks, Bacon led a rebellion. Bacon and thousands of disgruntled farmers led a revolt against Governor Berkeley and burned the capital.  This is one of several examples of social unrest within the colonies themselves.  

The momentum of a body is the product of its mass and velocity. In other words, P = m x v where P is the momentum of the body, m is its mass and v is its velocity. Since the units of mass are kg (typically) and those of velocity are m/s, the units of momentum are kg m/s. Here, the mass of the block is 2,000 g, which is same as 2 kg (since there are 1000 g in a kg) and the velocity of the block is 4 m/s. Thus, m = 2,000 g = 2 kg and v = 4 m/s Thus, P = m x v = 2 kg x 4 m/s = 8 kg m/s. Hence, the momentum of a block of mass 2 kg, rolling down a mountain side at 4 m/s is 8 kg m/s.  Note that we have assumed that the velocity of the block remains constant throughout its motion. Hope this helps.

It would seem that the United States caused the Cold War from the eight documents associated with this project.  The first document suggests a certain amount of naivete on the part of Franklin D. Roosevelt in trusting the intentions of Joseph Stalin for Eastern Europe after the war.  This miscalculation could infer that actions should have been made to prevent the Iron Curtain from descending on Eastern Europe after the war ended.  The third and sixth documents are written by the Soviet leadership.  They blame the United States and Britain for starting the war because of their goals of capitalist world domination.  The documents suggest that if weaker countries do not follow the lead of the Americans they will pay the price.  Even the sixth document, which was written by an American statesman is critical of the expansion of the American military.  In reading all of the documents, they seem to point to the United States as the instigator of the Cold War.  

When Romeo and Juliet meet at the Capulets' big party, one of the first things he compares her to is a "holy shrine," suggesting he is unworthy even to touch her, but that his lips are like "blushing pilgrims" that might help smooth the "profane" touch of his hands with a kiss (I.5.104-106). Romeo implies Juliet is divine, imbued with some force that is higher than human, and that he feels undeserving of her. Along these same lines, Romeo later calls Juliet a "saint" (I.5.114). Later, when Romeo speaks to Juliet on her balcony, he again calls her a "dear saint" (II.2.60). When Juliet expresses her fear that her kinsmen will find Romeo and kill him, he says, Alack, there lies more peril in thine eye Than twenty of their swords. Look thou but sweet, And I am proof against their enmity (2.2.76-78). In other words, then, Romeo assures Juliet that she has far more power over him than any weapons do. Romeo claims that, if Juliet cares abo...

You should analyze song lyrics in the same way that you would analyze a poem. As for the music itself, you need to identify the genre, which is pop. The period is simply when it was created. This comes from Lennon's later career, when he worked solo and with his wife, Yoko Ono. He recorded it in 1980, shortly before he was assassinated, but it was released posthumously in 1984. So the period of the song is the early 1980s, and like many songs of its time, it has a synthesized pop, new-wave sound, with a little bit of psychedelic funk thrown in.  The lyrics themselves are pretty straightforward, so you should first identify the main theme, which is about growing older, and how aging has brought clarity and greater happiness to the writer. Lennon equates being younger with psychological pain, confusion and delusion. The reason that the writer is happier now that he is older is that he has come to realize something very basic but important. This realization comprises his chorus, which...

When you begin writing the introduction of your paper on how you wish to become a journalist to fix the current issues of inaccuracies in the media, it might be a good idea to start with a quote from none other than Jon Stewart. Although he has never been a journalist, Stewart reached the zenith of journalism goals, which is something true journalists may not always be able to achieve. He once said, "Journalism is dead," a phrase that has been uttered before, but weighs more coming from him. This phrase causes shock. Make sure to quote the speaker. If you start your argument with that statement and cite Jon Stewart, you will pique readers' interest. They will wonder:  Why is the author quoting Jon Stewart? Does she/ he believe in that quote? What does the quote have to do with the essay? To support the use of the quote, you may want to write a small yet powerful reflection on what that quote means, and why the fact that it came from Jon Stewart makes it more controversial...

The Europeans had several motives during the Age of Exploration. One significant motive was to try to discover a shorter trade route to Asia. The Europeans were very interested in trading in Asia. The trip was a difficult one, and they hoped to find a new route that would take less time. The Europeans were also looking for new markets so they could expand their trade. The Europeans knew that if they could find new places with which to trade, they would be able to see economic growth in their countries. The Europeans also hoped to find minerals. If they could discover gold, silver, or other minerals, these discoveries would enhance their economies and their treasuries. Some countries also hoped to spread their religion. For example, Spain wanted to spread Christianity. If they could find new lands, they could convert more people to Christianity. The Europeans had several reasons for wanting to discover new lands.