Convergence of $\prod\limits_{k=1}^n \left(1+\frac{1}{k^2}\right) $ How can you determine whether the limit
$$ \displaystyle \lim_{n\to\infty} \prod_{k=1}^n \left(1+\frac{1}{k^2}\right)  $$ 
exists? Thanks for any hints!  

When checking convergence of infinite products of factors of the form $1 + \text{something positive}$, the inequalities
$$ 1 + \sum_{k=1}^n a_k \le \prod_{k=1}^n (1+a_k) \le \exp \sum_{k=1}^n a_k $$
are useful.  The lower one shows that if the infinite sum diverges, the infinite product does too; the upper one shows that if the infinite sum converges, the infinite product does too.  (To prove the lower inequality, imitate the proof of Bernoulli's inequality, which is a special case; to prove the upper, use $1+x\le e^x$.)
In your case, the corresponding sum is $\sum\limits_{k=1}^\infty \frac1{k^2}$, whose convergence can be checked by $p$-test, or by integral test, or by comparison to the telescoping sum $$1+\sum_{k=2}^\infty \frac1{k(k-1)} = 1+\sum_{k=2}^\infty \left(\frac1{k-1} - \frac1k\right),$$ or by recalling the result that made Euler famous, his solution to the Basel problem.
