I know that the composition of 2 analytic functions is analytic, and I conjectured that the radius of convergence of the resulting power series should be the smallest of the 2, but I'm not sure and I wouldn't know how to prove that.
Any help will be much appreciated. Thank you
Consider $$ \cosh z = \sum\limits_{n = 0}^\infty {\frac{{z^{2n} }}{{(2n)!}}} $$ and $$ \sqrt {1 + z} = \sum\limits_{n = 0}^\infty {\binom{1/2}{n}z^n } . $$ Tha radii of convergence is $\infty$ and $1$, respectively. The composition $$ \cosh \sqrt {1 + z} = \sum\limits_{n = 0}^\infty {\frac{{(1 + z)^n }}{{(2n)!}}} $$ is an entire function, since the series on the right-hand side converges for all complex $z$. Thus the radius of convergence of the power series of $\cosh \sqrt {1 + z}$ at the origin is $+\infty$ and not $1$ as it supposed to be based on your conjecture.