Does the following series absolutely converge, conditionally converge or diverge?
$$\sum_{n=1}^{+\infty}\frac{1}{n}\,\cos\left(\frac{\pi n}{2}\right)$$
My answer:
$$ 0<\frac{1}{\sqrt{n}}<\left|\frac{\cos(\pi n/2)}{n}\right| $$
and $\sum_{n\geq 1}\frac{1}{\sqrt{n}}$ diverges by p-series test so by comparison test, the original series must also diverge.
Best Answer
This series is conditionally convergent. Its terms are: $$\frac{1}{n}\cos\left(\frac{n\pi}{2}\right) = \left\{ \begin{array}{llll} \frac{1}{n} & \mbox{if } n=4k , k>0\\ 0 & \mbox{if } n=4k+1 , k\ge0\\ -\frac{1}{n} & \mbox{if } n=4k+2 , k\ge0\\ 0& \mbox{if } n=4k+3, k\ge0\\ \end{array} \right. $$ By removing zero expression, and just considering $n=4k$ and $n=4k+2$, we could replace and change indices. By write some sentences for this series: $$\sum_{n=1}^{\infty}\frac{1}{n}\cos\left(\frac{n\pi}{2}\right)=0-\frac{1}{2}+0+\frac{1}{4}+0-\frac{1}{6}+0+\frac{1}{8}+0+...=\\-\frac{1}{2}\Big(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...\Big)=-\frac{\ln{2}}{2}\approx-0.3466$$
It's conditionally convergences because $$\sum \left|\frac{\cos\left(\frac{n\pi}{2}\right)}{n}\right|=\frac{1}{2}\Big(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...\Big)$$ doesn't converge.
References: https://en.wikipedia.org/wiki/Alternating_series_test