Prove that $\prod_{k=1}^n (1+ \frac nk(e^{-i/n} -1))(1+ i/k)$ converges towards $\prod_{k=1}^\infty (1+1/k^2)$

Question

I look for an elegant proof that $$\prod_{k=1}^n \left(1+ \frac nk(e^{-i/n} -1)\right)\left(1+ \frac{i}k\right)$$ converges towards $\displaystyle \prod_{k=1}^\infty \left(1+\frac{1}{k^2}\right)$.

Here, $i^2 = -1$.

Answer 1

Elegant, not sure at all !

Let $$a_k= \Big[1+\frac{ n}{k}\left(e^{-\frac{i}{n}}-1\right)\Big]\left(1+\frac{i}{k}\right)$$ and you look for $$P_n=\prod_{k=1}^n a_k $$ When $n$ is large, using $$e^{-\frac{i}{n}}=1-\frac{i}{n}-\frac{1}{2 n^2}+O\left(\frac{1}{n^3}\right)$$ which makes $$a_k =\left(1+\frac{1}{k^2}\right)-\frac{k+i}{2 k^2 n}+O\left(\frac{1}{n^2}\right)$$