Is the sequence $a_n=\cos(n)/\sqrt{n+1}$ in $L_2$?

Question

I have been trying to show that the sequence $a_{n}$ belongs to $L_2$, $$ a_{n} =\frac{ \cos(n)} {\sqrt{n+1} } $$ But none of the test of convergence that i have tried, show that the following series converges. $$\sum_{n=1} ^{\infty} |a_{n} |^{2}$$

I wanted to ask for a hint for proving that the sequence does belong to $L_2$ space. Or perhaps it does not?

Answer 1

Write $\cos^2n$ as $\cos(2n)/2+1/2$; the series $\sum_{n\geqslant 1}\cos(2n)/(n+1)$ is convergent (use Abel's transform) and the series $\sum_{n\geqslant 1}1/(n+1)$ is divergent hence $(a_n)$ does not belong to $\ell^2$.

Answer 2

I think this will work: Since $\cos(n)$ will be equidistributed in $[-1,1]$ this means that $\cos^2(n)\geq \epsilon>0$ will hold for a positive fraction (depending on $\epsilon$)of indices $n$.

So you can lowerbound the sum by the Harmonic series.