summation algebra for $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1}$ Why does $\sum_{n=0}^\infty x^n + \sum_{n=0}^\infty x^{n+1} = 1 + 2\sum_{n=1}^\infty x^n$? Shouldn't this be $1 + x + 2\sum_{n=1}^\infty x^n$ because of the $n+1$ in the second summation?

This may convince you.
\begin{array}\
\displaystyle\sum_{n=0}^{\infty} x^n \hphantom{+ \displaystyle\sum_{n=0}^{\infty} x^{n+1}} &= 1 + \hphantom{2}x + \hphantom{2}x^2 + \dots\\
&\\
\hphantom{\displaystyle\sum_{n=0}^{\infty} x^n +} \displaystyle\sum_{n=0}^{\infty} x^{n+1} &= \hphantom{1 + 2}x + \hphantom{2}x^2 + \dots\\
&\\
\displaystyle\sum_{n=0}^{\infty} x^n + \sum_{n=0}^{\infty} x^{n+1} &= 1 + 2x + 2x^2 + \dots
\end{array}
