I was studying Taylor theorem when I came across this question in one of my math text books
Obtain Taylor's series expansion of the function $\sin(\frac {\pi}{4}+h)$ in ascending powers of $h$.
The first line of the solution was given as
The Taylor's expansion of $f(x+h)$ in powers of $h$ is
$$f(x+h)=f(x)+\sum_{n=1}^\infty \frac{h^n}{n!}f^{n}(x)$$
I have no idea how this came about.
Best Answer
The Taylor series of a function centered at some $x$ is given by $$ f(y)=\sum_{n=0}^\infty \frac{f^{(n)}(x)}{n!}(y-x)^n. $$ Now substitute in $y=x+h$ to get $$ f(x+h)=\sum_{n=0}^\infty \frac{f^{(n)}(x)}{n!}(x+h-x)^n. $$ Since $x+h-x=h$ and, for $n=0$ $$ \frac{f^{(0)}(x)}{0!}=f(x) $$ you get what you want.