Notes

To determine the interval of convergence for the given series: `sum_(n=1)^oo(-1)^nx^n/n` , we may apply Root Test. In Root test , we determine the limit as: `lim_(n-gtoo) root(n)(|a_n|)=L` or  `lim_(n-gtoo) |a_n|^(1/n)=L` The series is   absolutely convergent if it satisfies the Root test condition: L` lt1.` For the  given series: `sum_(n=1)^oo(-1)^nx^n/n` , we have: `a_n= (-1)^nx^n/n`  Then, set-up the limit as: `lim_(n-gtoo) |(-1)^nx^n/n|^(1/n) =lim_(n-gtoo) |x^n/n|^(1/n)` Note: `|(-1)^n|=1` and `1*|x^n| =|x^n|` . Apply Law of exponents: `(x/y)^n =x^n/y^n` and `(x^n)^m= x^(n*m).` `lim_(n-gtoo) |x^n/n|^(1/n)=lim_(n-gtoo) |(x^n)^(1/n)/n^(1/n)|`                         `=lim_(n-gtoo) |x^(n*1/n)/n^(1/n)|`                         `=lim_(n-gtoo) |x^(n/n)/n^(1/n)|`                          `=lim_(n-gtoo) |x^1/n^(1/n)|`                          `=lim_(n-gtoo) |x/n^(1/n)|` Evaluate the limit. `lim_(n-gtoo) |x/n^(1/n)| =( lim_(n-gtoo) |x|)/( lim_(n-gtoo) |n^(1/n)|)`                      `= |x...

The theme of "The Way Up to Heaven" is essentially revenge. Mr. Foster has been secretly tormenting his wife for years with his procrastination and deliberate delays. She doesn't resent his behavior because she can't be sure whether he is doing it intentionally or it is just an annoying character trait. One morning, she find definite evidence that he is a sadist who enjoys torturing her because he knows how important it is to her to be on time for appointments. Since Mr. Foster uses passive aggression to torment his wife, it is poetic justice that she should get revenge through the same passive aggression. She doesn't actively do anything to kill her husband; she only refrains from doing something. When Mr. Foster delays their trip to the airport by pretending he left something behind in their house, Mrs. Foster, waiting in the car, hears a sound of the elevator getting stuck between the floors. Instead of helping him, she proceeds to the airport, thereby letting ...

Dramatic irony occurs when the reader knows or understands something about the events that are unfolding that the characters do not.  This can often be tragic, such as in Shakespeare's Romeo and Juliet when the audience knows that Juliet is only asleep, but Romeo takes his own life because he believes that she is dead.  However, it can also be used for comedy, as it is in this story.  The narrator, although he is writing as an older man reflecting on his past experiences, speaks with a child's understanding of life and family.  When his mother says that she cannot afford a baby, he takes her literally, imagining that different babies come at different costs.  He thinks that his mother must be holding out for an expensive baby, saying, "The Geneys up the road had a baby, and everyone knew they couldn't afford seventeen and six.  It was probably a cheap baby, and Mother wanted something really good, but I felt she was too exclusive.  The Geneys' baby would have done...

Isabel takes three things from Miss Finch's home.  She takes her blankets, her shoes, and some flower seeds that her mother had collected.   Isabel's mother had collected the flower seeds and put them in a jar, but Isabel doesn't know what they will grow into.   At this point in the story, Mr. Robert is taking Isabel and Ruth to be sold, but he allows the two girls to go back into the house to collect their blankets and shoes.  He's not doing it to be nice either.  Having shoes and blankets will increase the slave value of Isabel and Ruth, and Mr. Robert is greedy.  Mr. Robert will  not allow Isabel to take anything else, so she needs to grab something that is easily transported and easily hidden.  The seeds fill both of those requirements and have the added sentimental value of reminding Isabel of her mother.   I looked around our small room, searching for a tiny piece of home I could hide in my pocket.  What to take? Seeds. On the hearth stood the jar of flower seeds...

Fossils are remnants of plants and animals that once lived. When they died, some of them had their bodies buried in the soil or water. Over time, deposition caused the formation of rocks, and these dead plants and animals were fossilized and preserved in these rocks. When we dig through rock layers, we sometimes come across fossils and scientists can use them to determine the age of the rocks they were buried in. The age of a fossil is determined by radiometric dating. The method uses the knowledge of half lives of radioactive isotopes of elements and the presence of these isotopes in animal/plant bodies. One such isotope is carbon-14, which has a half life of 5,730 years. This obviously cannot be used with Jurassic fossils or rocks, since the Jurassic period existed between 150-200 million years ago. For such samples, we can use Uranium-235, which has a half-life of about 700 million years and it decays to lead-207. Thus, knowing the ratio of U-235 and Pb-207, we can determine the age...

`a_n=nsin(1/n)` Apply n'th term test for divergence, which states that,  If `lim_(n->oo) a_n!=0` , then `sum_(n=1)^ooa_n` diverges `lim_(n->oo)nsin(1/n)=lim_(n->oo)sin(1/n)/(1/n)`   Apply L'Hospital's rule, Test L'Hospital condition:`0/0` `=lim_(n->oo)(d/(dn)sin(1/n))/(d/(dn)1/n)`  `=lim_(n->oo)(cos(1/n)(-n^(-2)))/(-n^(-2))` `=lim_(n->oo)cos(1/n)` `lim_(n->oo)1/n=0` `lim_(u->0)cos(u)=1`  By the limit chain rule, `` `=1!=0`  So, by the divergence test criteria series diverges.   ` `

`int1/(x^2-4x+9)dx` Let's complete the square for the denominator of the integral as: `(x^2-4x+9)=(x-2)^2+5`   `(x-2)^2+(sqrt(5))^2` `int1/(x^2-4x+9)dx=int1/((x-2)^2+(sqrt(5))^2)dx` Let's apply the integral substitution, substitute `u=x-2` `du=1dx` `=int1/(u^2+(sqrt(5))^2)du` Now use the standard integral :`int1/(x^2+a^2)=1/aarctan(x/a)` `=1/sqrt(5)arctan(u/sqrt(5))` substitute back u=(x-2) and add a constant C to the solution, `=1/sqrt(5)arctan((x-2)/sqrt(5))+C`