Alternating Series Remainder Theorem

Alternating Series Remainder Theorem

Read each question carefully and provide detailed solutions. Show all your work for full credit. Use the Alternating Series Remainder Theorem where applicable.

1. State the conditions under which an alternating series converges. What is the significance of the Alternating Series Remainder Theorem?

2. For the alternating series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n^2}$, what is the maximum error if we approximate the sum using the first 4 terms?

3. According to the Alternating Series Remainder Theorem, if $S$ is the sum of an alternating series satisfying the conditions for convergence, and $S_N$ is the sum of the first $N$ terms, then the absolute value of the remainder $|R_N| = |S - S_N|$ is less than or equal to the absolute value of the first   term, $a_{N+1}$.

4. Consider the series $\sum_{n=1}^{\infty} (-1)^{n} \frac{1}{n!}$. Approximate the sum of this series with an error less than $0.001$. Show your work.

5. The Alternating Series Remainder Theorem can be applied to any alternating series, regardless of whether it converges or diverges.

6. For the series $\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{\sqrt{n}}$, determine the number of terms needed to approximate the sum with an error less than $0.1$.