Why is the series expansion of $\sin(x)$ at $x=n\pi$ different to the expansion of $\sin(2x)$ at $x=\frac{n\pi}{2}?$

Question

I'm trying to do a series expansion of $\sin(2x)$ about the point $x=\frac{n\pi}2$ where is an integer.

I thought that the expansion would be the same as for $\sin(x)$ about the point $x=n\pi$ but I'm getting two different results for the different expansions. Why is this?

Answer 1

Recall that the graph of $\sin(2x)$ is that of $\sin(x)$ dilated parallel to the $x$-axis with scale factor $2$ (i.e. stretched out). Because of the stretching out, the expansion won't be the same as changing $x$ by some amount $\delta x$ in $\sin(2x)$ isn't the same as changing $x$ by $\delta x$ in $\sin(x)$.