Alternating Series Test — Question 4

*Question 4 Consider $\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt n}$.

1. Verify the AST hypotheses and classify absolute versus conditional convergence.

2. Find the smallest $N$ for which the next-term bound $b_{N+1}$ is strictly below $0.01$, thereby guaranteeing $|S-s_N|<0.01$.

3. Explain the role of the strict inequality.

*Question 4 – Solution

With $b_n=1/\sqrt n$, positivity is clear, $b_n$ decreases, and $b_n\to0$. The AST proves convergence. The absolute series $\sum1/\sqrt n$ diverges ($p=1/2$), so convergence is conditional.

$$|S-s_N|\le b_{N+1}=\frac1{\sqrt{N+1}}.$$ To guarantee an error strictly below $0.01$ it is sufficient that $$\frac1{\sqrt{N+1}}<\frac1{100} \iff N+1>10000.$$ Thus the smallest integer furnished by this bound is $\boxed{N=10000}$.

At $N=9999$, the bound equals $0.01$, not less than it. At $N=10000$, it is $1/\sqrt{10001}<0.01$.