Hello I have some problems concerning Taylor series.
Given the function $$f(x)=e^{\sin{x}} $$
I concluded that the Taylor series expansion would be
$$f(x) = \sum^\infty_{n=0}\frac{1}{n!}f^{(n)}(x)(x-x_0)^n$$ But Wolfram wrote a much simpler form
$$f(x)=\sum^\infty_{n=0}\frac{\sin^k{x}}{k!}$$
My question is: how come? Secondly, could somebody explain to me how to get the second sum's radius of convergence?
Thank you very much for your time!