Alternating Series Test — Question 10

*Question 10 Consider $\displaystyle\sum_{n=1}^{\infty}(-1)^n\frac{n}{n^2+1}$.

1. Use a derivative to verify decreasing magnitudes.

2. Apply the AST.

3. Test absolute convergence and classify the series.

*Question 10 – Solution

Let $f(x)=x/(x^2+1)$. Then $$f'(x)=\frac{1-x^2}{(x^2+1)^2}\le0\qquad(x\ge1).$$ Thus $b_n=n/(n^2+1)$ is nonincreasing and positive.

$$b_n=\frac{1/n}{1+1/n^2}\longrightarrow0.$$ Therefore the alternating series converges.

$$\lim_{n\to\infty}\frac{n/(n^2+1)}{1/n} =\lim_{n\to\infty}\frac{n^2}{n^2+1}=1.$$ By comparison with the divergent harmonic series, the absolute series diverges. Hence convergence is conditional.