I am trying to solve an exercise, but i am not sure that the result i get at the end is correct...Can I kindly ask you for a little help or a remark?
Find the radius of convergence of the following power series: $\sum_{n=0}^{\infty }(n^{2}+a^{n})z^{n}$ for any $z,a\in \mathbb{C}$
I use the quotient ratio: $\lim_{n\rightarrow \infty }\left | \frac{a_{n+1}}{a_n} \right |=\lim_{n\rightarrow \infty } \left | \frac{n^{2}+2n+1+aa^{n}}{n^{2}+a^{n}} \right |=\lim_{n\rightarrow \infty }\left | \frac{1+\frac{2}{n}+\frac{1}{n^{2}}+ \frac{aa^{n}}{n^{2}}}{1+\frac{a^{n}}{n^{2}}} \right | = 1 $ and then i get radius of convergence $1$.
With Cauchy-Hadamard I get $\limsup_{n\rightarrow \infty }\sqrt{n^{2}+a^{n}}=\limsup_{n\rightarrow \infty }\sqrt{n^{2}(1+\frac{a^{n}}{n^{2}})}=\limsup_{n\rightarrow \infty }n\sqrt{1+\frac{a^{n}}{n^{2}}}=\infty $ and a radius of convergence $0$.
I am not sure which result is correct if any of them is correct...
Thank you in advance!