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Determining whether $\sum_{n=0}^\infty \frac{\sin\left(\frac1{2^n}\right)}{\frac1{2^n}}$ is divergent.

I’m trying to determine whether the series
$$\sum_{n=0}^\infty \frac{\sin\left(\frac1{2^n}\right)}{\frac1{2^n}}$$
is divergent.
My idea is to consider
$$\sum_{n=0}^\infty \frac{\sin\left(\frac1{2^n}\right)}{\frac1{2^n}}\frac1{2^n}$$
and use the fact that $\sin(t)\geq \frac12|t|$, but I don’t know how to use this information.

A:

\begin{align} \sum_{n=0}^\infty \frac{\sin\left(\frac1{2^n}\right)}{\frac1{2^n}}\frac1{2^n} &=\sum_{n=0}^\infty \frac1{2^{n+1}}\frac1{2^n\sin(\frac1{2^n})}\\ &=\sum_{n=0}^\infty \frac1{2^{n+1}}\left(1-\frac1{2^{2n}}\right)^{ -1}\\ &=\sum_{n=0}^\infty \left(1-\frac1{2^{2n+1}}\right)^{ -1} \end{align}
This can be written as


Aug 31, 2013
Super Phisher is an excellent and simple tool to generate pishing sites.. genuine windows 7 ultimate download free, windows server 2008 .
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Aug 31, 2013
Super Phisher is an excellent and simple tool to generate pishing sites.. genuine windows 7 ultimate download free, windows server 2008 .