Exercise 7.35: Expressing some functions as series

Find a function representing each of the following series.

a) $\displaystyle \sum_{k=0}^{\infty} \frac{z^{2k}}{k!}$

b) $\displaystyle \sum_{k=1}^{\infty} k(z-1)^{k-1}$

c) $\displaystyle \sum_{k=1}^{\infty} k(k-1)z^{k-1}$

I'll try part (a). We know
$$e^{z} = \sum_{k=0}^{\infty} \frac{z^{k}}{k!},$$

So if we plug $z^2$ in for $z$, we get
$$e^{z^2} = \sum_{k=0}^{\infty} \frac{(z^2)^{k}}{k!} = \sum_{k=0}^{\infty} \frac{z^{2k}}{k!}.$$

So $e^{z^2}$ should work.

We know that $\frac{1}{1-z}=\sum_{k=0}^{\infty}z^k$ and want to find a function representing $\sum_{k=0}^{\infty}k(z-1)^{k-1}.$

So, $$\frac{1}{1-z}=\sum_{k=0}^{\infty}z^k$$
$$\frac{1}{(1-z)^2}=\sum_{k=0}^{\infty}kz^{k-1}$$ and
$$f(z)=\frac{1}{(1-(z-1))^2}=\sum_{k=0}^{\infty}k(z-1)^{k-1}$$