# How do you test the series sum_(n=1)^oo n/(n^2+2) for convergence?

Nov 19, 2017

This series diverges by the integral test.

#### Explanation:

Note that:

$x = \frac{1}{2} \frac{d}{\mathrm{dx}} \left({x}^{2} + 2\right)$

So:

$\int \frac{x}{{x}^{2} + 2} \mathrm{dx} = \frac{1}{2} \ln \left\mid {x}^{2} + 2 \right\mid + C \to \infty$ as $x \to \infty$

So ${\sum}_{n = 1}^{\infty} \frac{n}{{n}^{2} + 2}$ diverges by the integral test.

Alternatively, note that:

${\sum}_{n = 1}^{\infty} \frac{n}{{n}^{2} + 2} = {\sum}_{n = 1}^{\infty} \frac{1}{n + \frac{2}{n}} \ge {\sum}_{n = 1}^{\infty} \frac{1}{n + 2} = {\sum}_{n = 3}^{\infty} \frac{1}{n}$

which diverges.