# How do you test the series Sigma n/((n+1)(n^2+1)) from n is [0,oo) for convergence?

Jan 18, 2017

The series:

${\sum}_{n = 0}^{\infty} \frac{n}{\left(n + 1\right) \left({n}^{2} + 1\right)}$

is convergent.

#### Explanation:

You can test it by direct comparison, considering that for $n > 1$:

$0 < \frac{n}{\left(n + 1\right) \left({n}^{2} + 1\right)} < \frac{n}{n} ^ 3 = \frac{1}{n} ^ 2$

As the series:

${\sum}_{n = 0}^{\infty} \frac{1}{n} ^ 2$

is convergent based on the p-series test, then also the series:

${\sum}_{n = 0}^{\infty} \frac{n}{\left(n + 1\right) \left({n}^{2} + 1\right)}$

is convergent.