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Convergence of the infinite product $\prod_{n=1}^{\infty}\frac{1}{n^{2}+1}$

Does this product converge? $$\prod_{n=1}^{\infty}\frac{1}{n^{2}+1}$$ any hint?
Limit of $\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^3}\right)\dots \left(1-\frac{1}{n^n}\right)$ as $n\to \infty$
So I'm trying to solve the following limit: $$\lim_{n \to \infty}\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^3}\right)\dots \left(1-\frac{1}{n^n}\right)$$ Now, I tried getting the squeeze ...
how to prove that this sequence converges to $0$?
$0 \le x_0 \le \frac{1}{2}$ , and $x_{n+1}=x_n-\dfrac{4x_n^3}{n+1}$ When I take $x_0=\sqrt{\frac{1}{12}}$, it converges very very slow. I can see it is monotonic decreasing but don't know how to ...