Absolute Convergence and Divergence — Question 1

*Question 1 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{n^{3/2}}$.

1. Test the absolute-value series first.

2. Classify the original series and identify the strongest efficient theorem.

3. Give an upper bound for the absolute tail after $N$ terms.

*Question 1 – Solution

$$\sum\left|\frac{(-1)^n}{n^{3/2}}\right|=\sum\frac1{n^{3/2}}.$$ This is a convergent $p$-series because $p=3/2>1$.

Absolute convergence implies ordinary convergence, so the original series converges absolutely. Although the Alternating Series Test also proves ordinary convergence, it gives a weaker conclusion and is unnecessary here.

Since $x^{-3/2}$ is positive and decreasing, $$\sum_{n=N+1}^{\infty}\frac1{n^{3/2}}\le\int_N^{\infty}x^{-3/2}\,dx=\frac2{\sqrt N}.$$ This also bounds the signed remainder in magnitude.