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 .
