# Test the following series for convergence

Test the convergence of the following series:

$$\sum\frac{1}{n^2(1+\frac{1}{2}sin\frac{n\pi}{4})}$$

I have tried by ratio test,i.e.,$lim\frac{a_{n+1}}{a_n}=l$,then $\sum u_n$ will be convergent if $l<1$.But nothing can't be said from the form of the ratio I am getting.May be,my approach is wrong.

Note that since $-1\le \sin(n\pi/4)\le 1$ for all $n$, then

$$\frac23\le \frac{1}{1+\frac12 \sin(n\pi/4)}\le 2$$

Therefore, we find that

$$\frac23 \sum_{n=1}^N \frac1{n^2}\le \sum_{n=1}^N \frac1{n^2\left(1+\frac12 \sin(n\pi/4)\right)}\le 2 \sum_{n=1}^N \frac1{n^2}$$

Can you finish now?

• Yes,I can.Thank you :) – P.B. Jun 4 '16 at 15:22
• Pleased to hear! And you're quite welcome. My pleasure. -Mark – Mark Viola Jun 4 '16 at 15:25

hint:$$\sum\frac{1}{n^2(1+\frac{1}{2}\sin\frac{n\pi}{4})}\leq 2\sum\frac{1}{n^2}$$

• That inequality does not hold. – Mark Viola Jun 4 '16 at 15:14