Alternating Series Test — Question 2

*Question 2 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n^2}$.

1. Verify that the Alternating Series Test applies.

2. Determine whether the series converges absolutely or conditionally.

3. Give an error bound for $s_N$ and explain which conclusion is stronger.

*Question 2 – Solution

For $b_n=1/n^2$, we have $b_n>0$, $b_{n+1}<b_n$, and $b_n\to0$. Thus the alternating series converges.

$$\sum\left|\frac{(-1)^n}{n^2}\right|=\sum\frac1{n^2},$$ a convergent $p$-series with $p=2$. Hence the original series converges absolutely, a stronger conclusion than the AST alone provides.

$$|S-s_N|\le\frac1{(N+1)^2}.$$ In fact $S=-\pi^2/12$, though this exact value is not needed for classification.