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...