Strategy for Series — Question 3

*Question 3 Classify $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}$ as absolutely convergent, conditionally convergent, or divergent. Use separate tests for ordinary and absolute convergence.

*Question 3 – Solution

Put $b_n=1/\sqrt n$. Then $b_n>0$, $b_{n+1}<b_n$, and $$\lim_{n\to\infty}b_n=0.$$ Therefore $\sum(-1)^n b_n$ converges by the Alternating Series Test.

$$\sum_{n=1}^{\infty}\left|\frac{(-1)^n}{\sqrt n}\right|=\sum_{n=1}^{\infty}\frac1{n^{1/2}}.$$ This is a divergent $p$-series because $p=1/2\le1$.

The original series is conditionally convergent. One test establishes convergence; the second shows that convergence depends essentially on cancellation between alternating signs.