<s>[INST] How can I evaluate $\sum_{n=0}^\infty(n+1)x^n$? How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.
In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
 [/INST] If you want a solution that doesn't require derivatives or integrals, notice that
\begin{eqnarray}
1+2x+3x^2+4x^3+\dots = 1 + x + x^2 + x^3 + \dots \\
+ x + x^2+ x^3 + \dots\\
+ x^2 + x^3 + \dots \\
+x^3 + \dots \\
+ \dots \\
=1 + x + x^2 + x^3+\dots \\
+x(1+x+x^2+\dots) \\
+x^2(1+x+\dots)\\
+x^3(1+\dots)\\
+\dots \\
=(1+x+x^2+x^3+\dots)^2=\frac{1}{(1-x)^2}
\end{eqnarray}
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