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I am doing an exercise for which I have the sum below on the LHS and I am supposed to rearrange it to get the sum on the RHS. \begin{equation} z\sum\limits_{n=0}^{\infty}(n+1) \left( \frac{1}{z^{2n}} \right)=\frac{1}{z}+\sum_{n=1}^{\infty}(n+1)\left(\frac{1}{z^{2n+1}}\right) \end{equation} I can't seem to figure out how to get to the equation on the RHS. I did the following,

\begin{equation} z\sum\limits_{n=0}^{\infty}(n+1)(\frac{1}{z^{2n}})=\sum_{n=0}^{\infty}(n+1)z^{1-2n}=z+\sum\limits_{n=1}^{\infty}(n+1)z^{1-2n} \end{equation} I wanted to check whether these two answers are the same, so I did a lengthy computation to determine that \begin{equation} z+\sum\limits_{n=1}^{\infty}(n+1)z^{1-2n}=z+\frac{z(2z^{2}-1)}{(z^{2}-1)^{2}}=\frac{z^{5}}{(z^{2}-1)^{2}} \end{equation} Doing the same with the given answer I get, \begin{equation} \frac{1}{z}+\sum\limits_{n=1}^{\infty}(n+1)(\frac{1}{z^{2n+1}})=\frac{1}{z}+\frac{2z^{2}-1}{z(z^{2}-1)^{2}}=\frac{z^{3}}{(z^{2}-1)^{2}} \end{equation} The start of this exercise required me to compute that \begin{equation} \frac{z^{3}}{(z^{2}-1)^{2}}=z\sum\limits_{n=0}^{\infty}(n+1)(\frac{1}{z^{2n}}) \end{equation} so I know that the given answer should be correct. Somehow I can't seem to figure out where I go wrong in my approach, and what approach should be taken to determine the given answer.

David G. Stork
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TK99
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    I think your computations are correct and the two sides are not equal. Either the left-hand side should have $z$ replaced by $\frac{1}{z}$, or the right-hand side should have $\frac{1}{z^{2n-1}}$ instead of $\frac{1}{z^{2n+1}}$. – angryavian Aug 10 '21 at 20:27
  • Thank you, after confirming that my computations were indeed correct I decided to go back to the beginning and I realised there was a small mistake in my work. This did indeed give the $\frac{1}{z}$ instead of $z$. – TK99 Aug 10 '21 at 20:37

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If you claim the above to be true then (starting the RHS from $n=0$): $$\sum_{n=0}^\infty(n+1)z^{-(2n - 1)} = \sum_{n=0}^\infty(n+1)z^{-(2n+1)}$$ Which seems to be false (unless $z^2=1$)

Sergey Zaitsev
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