Let $f$ be a function that is known to be analytic in a neighbourhood of the origin in the complex plane. Furthermore, it is known that for $n ∈ \mathbb N$, $$f^{\left(n\right)}\left(0\right)=\left(n-1\right)!\left(n+1\right)\left(\frac{\left(n+1\right)}{n}\right)^{\left(n+1\right)\left(n-1\right)}$$
Find the radius of the largest circle with centre at the origin inside which the Taylor series of $f$ defines an analytic function.
My Try:-
I know that $\frac{1}{R}=\lim \sup|a_n|^{\frac{1}{n}}=\lim_{n\to \infty}\left(n-1\right)!^{\frac{1}{n}}\left(n+1\right)^{\frac{1}{n}}\left(\frac{\left(n+1\right)}{n}\right)^{\frac{\left(n+1\right)\left(n-1\right)}{n}}$ How do I find the limit of this sequence? I took logarithm. But the terms are becoming complicated.