<s>[INST] Sum of the series $\sum \frac{n}{2^{n}}$ I know that the series converges by d'Alembert ratio test, where $\lim\left ( \frac{A_{n+1}}{A_{n}} \right )= \frac{1}{2}$, but I don't know how to calculate the sum of the serie. Thanks for the help.
 [/INST] \begin{align*}
S&= \frac{1}{2}+\frac{2}{4}+\frac{3}{8}+\frac{4}{16}+.....\\
\frac{1}{2}S&=\frac{1}{4}+\frac{2}{8}+\frac{3}{16}\\
S-\frac{1}{2}S&=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+......\\
S-\frac{1}{2}S&=1
\end{align*}
Thus $S=2$
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