Notes

To determine the convergence or divergence of the series `sum_(n=1)^oo (-1)^(n+1)/(nsqrt(n))` , we may apply Alternating Series Test . In Alternating Series Test, the series `sum (-1)^(n+1) a_n ` is convergent if: 1) `a_ngt=0` 2) ` a_n` is monotone and decreasing sequence. 3) `lim_(n-gtoo) a_n =0` For the series `sum_(n=1)^oo (-1)^(n+1)/(nsqrt(n))` , we have: `a_n = 1/(nsqrt(n))` Apply the radical property: `sqrt(x) =x^(1/2)` and Law of Exponents: `x^n*x^m =x^(n+m).` `a_n = 1/(nsqrt(n))`       `=1/(n*n^(1/2))`       `=1/n^(1+1/2)`       `=1/n^(3/2)` The `a_n =1/n^(3/2) ` is a decreasing sequence. Then, we set-up the limit as : `lim_(n-gtoo)1/n^(3/2) = 1/oo =0` By alternating series test criteria, the series `sum_(n=1)^oo (-1)^(n+1)/(nsqrt(n))` converges . The series `sum_(n=1)^oo (-1)^(n+1)/(nsqrt(n))` has positive and negative elements . Thus, we must verify if the series converges absolutely or conditionally. Recall: a) Absolute Convergence :  `sum a_n`  is absolutely convergent if `...

The present day Republican Party is very different from the party of Abraham Lincoln as its core ideology has undergone radical change. The party was founded on the principle of opposition to the expansion of slavery. This was in reaction to the Kansas-Nebraska Act, which was designed to open up the Kansas and Nebraska territories to slavery. This would have gone against the Missouri Compromise, which forbade slavery north of 36°30'. During the Lincoln era, the Party stood for the abolition of slavery and high tariffs to protect American industry; it introduced the income tax and generally favored big government. The party was dominated by moderate Protestants from the Northern states, African Americans, and owners of big businesses. Later on, the party also had an expansive foreign policy under William McKinley and Theodore Roosevelt. The Republicans dominated government between the Lincoln era and 1932, when Franklin D. Roosevelt came to power under the Democratic Party. The Repu...

We are asked to find the subgroup of the group of integers modulo 8 under addition generated by the element 2: The elements of (Z8,+) are G={0,1,2,3,4,5,6,7} with 0 the identity element for the operation +. We can generate all of the subgroups using addition modulo 8: [0]={0} [1]={0,1,2,3,4,5,6,7}=G [2]={0,2,4,6} [3]={3,6,1,4,7,2,5,0}=G [4]={0,4} [5]={5,2,7,4,1,6,3,0}=G [6]={6,4,2,0}=[2] [7]={7,6,5,4,3,2,1,0}=G The subgroup generated by [2] is {0,2,4,6} Note that this is a subgroup: there is an identity {0}, it has the associative property as integer addition is associative, it has the closure property, and every element has an inverse. (0 is its own inverse, 2+6=6+2=0, and 4 is its own inverse.) Also note that the order ("size") of the subgroups are factors of 8, namely 1,2,4, and 8. You can also look at the gcf between the generating element and 8 and compare to the "size" of the generated subgroup.

To apply the  Root test  on a series `sum a_n` , we determine the limit as: `lim_(n-gtoo) root(n)(|a_n|)= L` or `lim_(n-gtoo) |a_n|^(1/n)= L` Then, we follow the conditions: a) `Llt1` then the series is  absolutely convergent . b) `Lgt1` then the series is  divergent . c) `L=1` or  does not exist   then the  test is inconclusive . The series may be divergent, conditionally convergent, or absolutely convergent. We may apply the  Root Test  to determine the convergence or divergence of the  series  `sum_(n=1)^oo (n/(2n+1))^n` . For the given series `sum_(n=1)^oo(n/(2n+1))^n` , we have `a_n =(n/(2n+1))^n.` Applying the Root test, we set-up the limit as:  `lim_(n-gtoo) |(n/(2n+1))^n|^(1/n) =lim_(n-gtoo) ((n/(2n+1))^n)^(1/n)`  Apply the Law of Exponents:`(x^n)^m= x^(n*m)` . `lim_(n-gtoo) ((n/(2n+1))^n)^(1/n) =lim_(n-gtoo) (n/(2n+1))^(n*(1/n) )`                                   `=lim_(n-gtoo) (n/(2n+1))^(n/n )`                                   `=lim_(n-gtoo) (n/(2n+1))^1`                  ...

Direct comparison test is applicable when `suma_n` and `sumb_n` are both positive series for all n such that `a_n<=b_n` If `b_n` converges ,then `a_n` converges. If `a_n` diverges, then `b_n` diverges. `sum_(n=0)^oo4^n/(5^n+3)` Let `a_n=4^n/(5^n+3)` and `b_n=4^n/5^n=(4/5)^n` `4^n/5^n>4^n/(5^n+3)>0`  for `n>=1` `sum_(n=0)^oo(4/5)^n` is a geometric series with ratio r`=4/5<1` A geometric series with ratio r , such that `|r|<1` converges. The geometric series `sum_(n=0)^oo(4/5)^n` converges,so the series `sum_(n=0)^oo4^n/(5^n+3)` converges as well , by the direct comparison test.

There are several things you will need to do the develop these draft paragraphs into an essay. First, you need a clear thesis statement. If you want to focus on the lack of name for the Little Convent Girl, you should make that a thesis, and then devote the rest of the essay to that issue. Otherwise, you need to start your introduction with a clear sense of your main points and how they are tied together.  You might argue that the center of the story "The Little Convent Girl" is really the discovery we make at the end that her mother is black, an important point you make in the first paragraph. You could then use as supporting evidence for the notion that what matters is the issue of identity in the way the girl is discussed. Her lack of a name, as you say, paradoxically emphasizes the issue of her identity. It makes the issue of her identity one that is foregrounded. The problem of identity is universalized by her lack of name. The word "girl" references the ways t...

Baldeo's death shows an ordinary man's extraordinary courage. Baldeo is quite ordinary.  There is nothing that outwardly distinguishes him from anyone else.  He works a standard job as the night watchman.  Yet, Bond infuses Baldeo with extraordinary courage when he has to come face to face with the tiger in the tunnel.  Baldeo shows incredible resolve as the "huge body of the tiger [is] trotting steadily towards him."  It would be completely understandable if Baldeo ran away. However, Baldeo's courage is evident in how he knows that "flight was useless."  He knows that the tiger is "more sure-footed" and could take him down if Baldeo retreated. Baldeo shows heroic stature, having "dared to stand in the way" while "the great brute [was] moving rapidly towards him."  As the tiger attacked, Baldeo uses his axe to defend himself.  Baldeo never once backs down, even when he has lost his weapon and the tiger "sprang upon him....