Question

Determine whether the following series converges absolutely, conditionally or diverges.

SUM (-1)^n * n/(n^2+1), from 1 to inf

Answer 1

\[\sum_{1}^{\inf} (-1)^{n} * n/(n ^{2} +1)\]

Answer 2

n=1 is suppose to be on the sum symbol.

Answer 3

I'm not quite sure where to start on this.

Answer 4

\( \frac n {n^2 +1} \) behaves like \( \frac 1 n\) near \(\infty \). Hence the series converges conditionally.

Answer 5

It satisfies the conditions of a convergent alternating series, but does not
converge absolutely, since the series of absolute values behave like the divergent harmonic series \[ \sum_{n=1}^\infty \frac 1 n \]