Here's a problem I'm stuck on:
Determine the power series and radius of convergence for $f(z) = \sin z$ centered at $1$.
I'm unsure as to what to do with the fact that it is centered about $1$ in terms of the power series. Any help would be appreciated. Thanks!
Best Answer
We know that $a_{n}=f^{(n)}(1)/n!$, and $|f^{(n)}(1)|$ is either $\sin 1$ or $\cos 1$. We know that $1/R=\limsup_{n}|a_{n}|^{1/n}\leq M\limsup_{n}1/(n!)^{1/n}=0$, so $R=\infty$. The fact that $(n!)^{1/n}\rightarrow\infty$ follows by $n!>(n/2)^{n/2}$ for large $n$.