# Convergence/Divergence of infinite series

I'm trying to figure out if the following series converges or diverges. I have spent hours on it and can't figure it out. Tried to use the comparison test, Dirichlet, and Abel, but none of it worked. Any hints?

$$\sum_{n=1}^\infty \frac{\cos^2(n)}{\sqrt{n}}$$

• Hey can you show what you did for the Abel comparasin test? – Max0815 Feb 6 at 19:38

HINT:

Recall that $$\cos^2(x)=\frac{1+\cos(2x)}{2}$$.

• Is using this and splitting the series into two helpful? Is the addition of a divergent and a convergent series also divergent? – peroxisome7 Feb 6 at 19:57
• @peroxisome7, Yes. That is a version of contrapositive statement of the fact that difference of two convergent series is again convergent. – Sangchul Lee Feb 6 at 20:07
• @peroxisome Note that $$\sum_{n=1}^N \frac{\cos^2(n)}{\sqrt n} =\frac12 \sum_{N=1}^N \frac{1}{\sqrt n}+\frac12 \sum_{n=1}^N \frac{\cos(2n)}{\sqrt n}$$Now, Dirichelt's test shows that the second partial sum on the right-hand side converges. If the partial sum on the left-hand side converged, then the first partial sum on the right-hand side would be the sum (difference) of two convergent partial sums and hence would also converge. Inasmuch as the first partial sum on the right-hand side diverges, what can one conclude? – Mark Viola Feb 6 at 21:20

Another approach is to observe that among $$e^{in}, n=1,2,\dots, 7,$$ at least one of these points lies in the arc $$\{e^{it}: t\in (-\pi/4,\pi/4)\}.$$ Thus

$$\sum_{n=1}^{7} \frac{\cos^2 n}{\sqrt n} \ge \frac{1}{ 2}\frac{1}{\sqrt 7}.$$

The same thing happens for $$n=8,\dots,14.$$ etc. So the series in question is at least

$$\sum_{m=1}^{\infty} \frac{1}{ 2}\frac{1}{\sqrt {7m}} = \infty.$$

Note that this idea will work if $$\cos^2 n$$ is replaced by $$|\cos n|^p$$ for any $$p>0$$.

• (+1) I remember having used this trick to solve a previous question on the site. Under its apparence of simplicity, it is indeed rather powerful. – Did Feb 6 at 20:05
• @zhw. Inasmuch as at least one of the points $1,2,\dots, 7$ lies in that arc, $\cos^2(n)\ge 1/2$ for such a point. Then, $\sum_{n=1}^7 \frac{\cos^2(n)}{\sqrt n}\ge \frac1{2\sqrt 7}$, would it not? – Mark Viola Feb 6 at 20:42
• @MarkViola Right you are. Now edited. – zhw. Feb 6 at 23:20