<s>[INST] Proof of sum results I was going through some of my notes when I found both these sums with their results
$$
x^0+x^1+x^2+x^3+... = \frac{1}{1-x}, |x|<1
$$
$$
0+1+2x+3x^2+4x^3+... = \frac{1}{(1-x)^2}
$$
I tried but I was unable to prove or confirm that these results are actually correct, could anyone please help me confirm whether these work or not?
 [/INST] Others have answered on how to justify that $S=1+x+x^2+\cdots+x^n+\cdots$ equals $\frac{1}{1-x}$ for $|x|<1$. To get the second identity without differentiation, note that
\begin{align*}
1+2x+3x^2+4x^3+\cdots&=1+(x+x)+(x^2+x^2+x^2)+(x^3+x^3+x^3+x^3)+\cdots\\
&=(1+x+x^2+x^3+\cdots)+(x+x^2+x^3+\cdots)+(x^2+x^3+\cdots)+\cdots\\
&=S+xS+x^2S+\cdots=S(1+x+x^2+\cdots)\\
&=S^2\\
&=\frac{1}{(1-x)^2}\cdot
\end{align*}
</s>