Summation by parts: Let $\{a_k\}$ and $\{b_k\}$ be arbitrary sequences of real numbers, and let $s_n=\sum_{k=0}^n a_k$. Then for $0\le m< n$,
$\sum\limits_{k=m}^na_kb_k=s_nb_n-s_{m-1}b_m+\sum\limits_{k=m}^{n-1}s_k(b_k-b_{k+1})$
The series $t=\sum_{k=1}^\infty (\sin k)/k$ converges. Use summation by parts to show with certainty that $t>0$.
I am not sure how to show this but here is my start:
Let $b_k=\frac{1}{k}$ and $a_k=\sin k$. Also $s_n=\sum_{k=0}^n\sin k$ for $n\in\mathbb{N}$.
Note $|\sum_{k=1}^n|\le\csc(1/2)$.
Then
$\sum\limits_{k=m}^n\frac{1}{k}\sin k=(\sum_{k=1}^n\sin k)\cdot\frac{1}{n}-(\sum_{k=1}^{m-1}\sin k)\cdot\frac{1}{m}+\sum_{k=m}^{n-1}(\sum_{j=1}^k\sin j)(\frac{1}{k}-\frac{1}{k+1})$
How can I introduce the $|\sum_{k=1}^n|\le\csc(1/2)$? and then use that to show that $t>0$?
