<s>[INST] What is the sum of $E=\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+...$ What is the right way to assess this problem?
$$E=\frac{1}{3}+\frac{2}{3^{2}}+\frac{3}{3^{3}}+\frac{4}{3^{4}}+...$$
To find the value of the sum I tried to use the fact that it could be something convergent like a geometric series. However it does not seem to be the case as there is no common ratio between the terms. Since the numerator is increasing from $1,2,3,...$ makes it impossible to find a ratio. What should I do?.
 [/INST] You can split it up into a sum of geometric progressions:
\begin{align} 3^{-1} + 2 \cdot 3^{-2} + 3 \cdot 3^{-3} + \dotsb = 3^{-1} + 3^{-2} + 3^{-3} + \dotsb \\
+ 3^{-2} + 3^{-3} + \dotsb \\
+ 3^{-3} + \dotsb \\
\ddots \\ 
= 1(3^{-1}+ 3^{-2} + 3^{-3} + \dotsb) \\
+3^{-1}(3^{-1}+ 3^{-2} + \dotsb) \\
+ 3^{-2}(3^{-1} + \dotsb) \\
\ddots \\
= (1+3^{-1}+3^{-2}+\dotsb)(3^{-1}+3^{-2}+3^{-3}+\dotsb)
\end{align}
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