Absolute Convergence and Divergence — Question 5

*Question 5 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}$.

1. Verify ordinary convergence.

2. Test the absolute-value series with the $p$-series rule.

3. Classify convergence and give an error bound.

*Question 5 – Solution

With $b_n=1/\sqrt n$, positivity, decrease, and $b_n\to0$ all hold. Thus the series converges.

$$\sum\left|\frac{(-1)^n}{\sqrt n}\right|=\sum\frac1{n^{1/2}},$$ which diverges because $p=1/2\le1$.

The convergence is conditional, and $$|S-s_N|\le\frac1{\sqrt{N+1}}.$$