What is the convergence (absolute or conditional) of the alternate series: f(x)=sum_{n=1} ^oo (-1)^n ((n)/(7n^2+3))  ?

Mar 9, 2018

This series is conditionally convergent but not absolutely convergent.

Explanation:

Since this series is alternating, it will conditionally converge provided the individual terms tend to $0$, which they do:

${\lim}_{n \to \infty} \left(\frac{n}{7 {n}^{2} + 3}\right) = {\lim}_{n \to \infty} \frac{1}{7 n + \frac{3}{n}} = 0$

On the other hand, it is not absolutely convergent, since when $n \ge 1$ we have:

$7 {n}^{2} + 3 \le 10 {n}^{2}$

So:

$\frac{n}{7 {n}^{2} + 3} \ge \frac{n}{10 {n}^{2}} = \frac{1}{10} \frac{1}{n}$

So:

${\sum}_{n = 1}^{\infty} \left\mid {\left(- 1\right)}^{n} \left(\frac{n}{7 {n}^{2} + 3}\right) \right\mid \ge \frac{1}{10} {\sum}_{n = 1}^{\infty} \frac{1}{n} = \infty$