Absolute Convergence and Divergence — Question 5

Question 5 – Solution

Step 1: Apply the AST.

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

Step 2: Test absolute convergence.

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

Step 3: Classify and estimate.

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