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How to find the sum of this infinite series
What is the limit of $\sum \limits_{n=1}^{\infty}n(2/3)^n$

I am wondering how to sum this infinite sequence: $\displaystyle\sum\limits_{k=1}^\infty \frac{k}{2^k}$. According to wolfram alpha this equals to $2$, but I would like to know why.

Thanks in advance!

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marked as duplicate by Jonas Meyer, J. M., David Mitra, Brian M. Scott, Srivatsan Dec 17 '11 at 12:32

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I'll never pass an opportunity to link: mathdl.maa.org/images/cms_upload/268948749035.pdf (which gives the value of ${1\over2}\sum_{k=1}^\infty {k\over2^k}$). –  David Mitra Dec 17 '11 at 12:27
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@David: I like $$\begin{array}{} \frac12&+&\frac14&+&\frac18&+&\frac1{16}&+&\dots&=&1\\ &&\frac14&+&\frac18&+&\frac1{16}&+&\dots&=&\frac12\\ &&&&\frac18&+&\frac1{16}&+&\dots&=&\frac14\\ &&&&&&\frac1{16}&+&\dots&=&\frac18\\ &&&&&&&&\vdots&&\vdots\\ \hline \frac12&+&\frac24&+&\frac38&+&\frac4{16}&+&\dots&=&2 \end{array}$$ –  Brian M. Scott Dec 17 '11 at 12:37