# Consider the series sum_{n=1}^inftyfrac{(-1)^n}{n^2} a) Show the series converges or diverges using the alternating series test. b) Approximate the sum using the 4-th partial sum(S_4) of the series. c) Calculate the maximum error between partial sum(S_4) and the sum of the series using the remainder term portion of the alternating series test.

Question
Series
Consider the series $$\sum_{n=1}^\infty\frac{(-1)^n}{n^2}$$
a) Show the series converges or diverges using the alternating series test.
b) Approximate the sum using the 4-th partial sum($$S_4$$) of the series.
c) Calculate the maximum error between partial sum($$S_4$$) and the sum of the series using the remainder term portion of the alternating series test.

2021-03-08
Given series is
$$\sum_{n=1}^\infty\frac{(-1)^n}{n^2}=\sum_{n=1}^\infty(-1)^nb_n$$
$$b_n=\frac{1}{n^2}$$
Since,
$$n^2<(n+1)^2\forall n\in N$$</span>
$$\frac{1}{n^2}>\frac{1}{(n+1)^2}\forall n\in N$$
$$b_n>b_{n+1}\forall n\in N$$
Therefore, sequence $$\left\{b_n\right\}$$ is decreasing sequence
Also,
$$\lim_{n\to\infty}b_n=\lim_{n\to\infty}\frac{1}{n^2}=0$$
Therefore, Alternating series test, given series converges.
b) 4th partial sum of the series is evaluated as follows
$$\sum_{n=1}^\infty a_n=\sum_{n=1}^\infty\frac{(-1)^n}{n^2}$$
$$a_n=\frac{(-1)^n}{n^2}$$
$$S_4=a_1+a_2+a_3+a_4$$
$$S_4=\frac{(-1)^1}{1^2}+\frac{(-1)^2}{2^2}+\frac{(-1)^3}{3^2}+\frac{(-1)^4}{4^2}$$
$$=-1+\frac{1}{4}-\frac{1}{9}+\frac{1}{16}=-0.7986$$
c) By Alternating series error estimation theorem,, Maximum error between $$S_4$$ and sum S of the series is
$$|S-S_4|\leq|a_5|$$
$$|S-S_4|\leq|\frac{(-1)^5}{5^2}|=0.04$$
Maximum error between $$S_4$$ and sum S of the series is 0.04

### Relevant Questions

Consider the telescoping series
$$\sum_{k=3}^\infty(\sqrt{k}-\sqrt{k-2})$$
a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series?
b) Write out the partial sums $$s_3,s_4,s_5$$ and $$s_6$$.
c) Compute the n-th partial sum $$s_n$$, and put it in closed form.
Consider the telescoping series
$$\displaystyle{\sum_{{{k}={3}}}^{\infty}}{\left(\sqrt{{{k}}}-\sqrt{{{k}-{2}}}\right)}$$
a) Apply the Divvergence Test to the series. Show all details. What conclusions, if any, can you make about the series?
b) Write out the partial sums $$\displaystyle{s}_{{3}},{s}_{{4}},{s}_{{5}}$$ and $$\displaystyle{s}_{{6}}$$.
c) Compute the n-th partial sum $$\displaystyle{s}_{{n}}$$, and put it in closed form.
Use Theorem Alternating Series remainder to determine the number of terms required to approximate the sum of the series with an error of less than 0.001.
$$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n^5}$$
Use Theorem Alternating Series remainder to determine the number of terms required to approximate the sum of the series with an error of less than 0.001.
$$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{2n^3-1}$$
Consider the following infinite series.
a.Find the first four partial sums $$S_1,S_2,S_3,$$ and $$S_4$$ of the series.
b.Find a formula for the nth partial sum $$S_n$$ of the indinite series.Use this formula to find the next four partial sums $$S_5,S_6,S_7$$ and $$S_8$$ of the infinite series.
c.Make a conjecture for the value of the series.
$$\sum_{k=1}^\infty\frac{2}{(2k-1)(2k+1)}$$
Determine whether the series converges or diverges and justify your answer. If the series converges, find its sum
$$\sum_{n=0}^\infty\frac{3^{n-2}}{4^{n+1}}$$
Find a formula for the nth partial sum of the series and use it to determine whether the series converges or diverges. If a series converges, find its sum.
$$\sum_{n=1}^\infty(\cos^{-1}(\frac{1}{n+1})-\cos^{-1}(\frac{1}{n+2}))$$
Determine whether the series $$\sum a_n$$ an converges or diverges: Use the Alternating Series Test.
$$\sum_{n=2}^\infty(-1)^n\frac{n}{\ln(n)}$$
$$\sum_{n=1}^\infty\frac{4^n}{n+1}$$
$$\sum_{n=2}^\infty\frac{1}{4^n}$$