Convergence of $P_{n}=\frac{1}{n^{2}}\prod_{k=1}^{n}{(n^{2}+k^{2})^{\frac{1}{n }}}$

Question

I got some issues with the convergence of Pn

$P_{n}=\frac{1}{n^{2}}\prod_{k=1}^{n}{(n^{2}+k^{2})^{\frac{1}{n }}}$ , with n⩾1

and finding the limit of this convergent sequence , thanks in advance .

Answer 1

Hint: $\ln P_n=\displaystyle\frac{1}{n}\sum_{k=1}^{n}\ln\Big(1+\frac{k^2}{n^2}\Big)$ is a Riemann sum for $\displaystyle\int_0^1\ln(1+x^2)\,dx$ (which is elementary).