Power Series defined incorrectly.

Question

The power series is defined as: $$\sum^\infty_{k=0}a_k(z-z_0)^k=f(z)$$Where $a_k$ are coefficients that make this true. But according to this definition, the sine function:$$\sum_{k\ge0}(-1)^k\frac{z^{2k+1}}{(2k+1)!}=\sin x$$ Couldn't be a power series. Notice that the power is $2k+1$ and not $k$. Is there something wrong in my understanding?

Answer 1

You can think of it as every other term being $0$. That is,

$$\sin(z) = \sum_{n=0}^\infty a_n z^n$$

where

$$a_n = \begin{cases} (-1)^k/(2k+1)!, & n = 2k+1 \text{ for some integer } k \\ 0, & \text{otherwise} \end{cases}$$