<s>[INST] Showing series expansion How would I show $$\frac{\sinh z}{z^2}=\frac{1}{z}+\sum_0^\infty \frac{z^{2n+1}}{(2n+3)!}$$ 
I have $\sinh z=\sum_0^\infty\frac{z^{2n+1}}{(2n+1)!}$ and if I multiply by $\frac{1}{z^2}$ then $\frac{1}{z^2}\sinh z=\sum_0^\infty \frac{1}{z^2}\frac{z^{2n+1}}{(2n+1)!}$ but I how would I would factor out $\frac{1}{z}$? 
 [/INST] Write out a few terms to see what is going on.
$$\begin{align}
\sinh z &= \frac{z}{1!}+\frac{z^3}{3!} + \frac{z^5}{5!}+\dots
\\
\frac{\sinh z}{z^2} &= \frac{1}{1! z}+\frac{z}{3!} + \frac{z^3}{5!}+\dots
\end{align}$$
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