Answer
$\displaystyle \frac{\sin 1}{1-\sin 1}\approx 5.308$
Work Step by Step
$\sin 1$ is constant , and this is a geometric series, with
$a=1$
$|r|=|\sin 1|\approx $0.841470984808 $ < 1$
so, by Th 9/6, the series $\displaystyle \sum_{n=0}^{\infty}ar^{n}$ converges,
$\displaystyle \sum_{n=0}^{\infty}ar^{n}=\frac{a}{1-r}$.
BUT, be careful,
the indexing of the sum in Th.9.6 starts with n=0, so we need to adjust:
$\displaystyle \sum_{n=0}^{\infty}r^{n}=1+r+r^{2}+r^{3}...$
$\displaystyle \sum_{n=1}^{\infty}r^{n}=r+r^{2}+r^{3}+r^{4}...$
so, $\displaystyle \sum_{n=1}^{\infty}r^{n}=r\cdot\sum_{n=0}^{\infty}r^{n}$
$\displaystyle \sum_{n=1}^{\infty}[\sin 1]^{n}=\sin(1)\sum_{n=0}^{\infty}[\sin 1]^{n}$
... apply the theorem on the sum...
$=\displaystyle \sin 1\cdot\frac{1}{1-\sin 1}$
$=\displaystyle \frac{\sin 1}{1-\sin 1}\approx 5.308$
