Question #373e2

1 Answer
Nov 26, 2017

This series is conditionally convergent.

Explanation:

The series sum_{n=0}^{infty}(-1)^{n}n/(n^2+1) converges by the Alternating Series Test (since its terms alternate in sign and since n/(n^2+1) forms a decreasing sequence that approaches 0 as n-> infty).

However, the corresponding series sum_{n=0}^{infty}|(-1)^{n}n/(n^2+1)|= sum_{n=0}^{infty}n/(n^2+1) diverges. This can be seen by noting that n/(n^2+1) geq 1/(2n)>0 for all integers n geq 1, that sum_{n=1}^{\infty}1/(2n) diverges (its terms are one-half the terms of the divergent harmonic series ), and then applying the Comparison Test .

The facts in the two paragraphs above mean that the original series sum_{n=0}^{infty}(-1)^{n}n/(n^2+1) converges conditionally .