Are there limits of which we're not able yet to find the value or not even prove non/existence? I really like working out limits, so I've been wondering:

Are there limits we're struggling to evaluate? Are there limits of which we're not succeeding in proving the existence or nonexistence?


I feel like this is cheating but it is well known that
\begin{align*}
\sum_{n=1}^{\infty} \frac{1}{n^{2}} = \frac{\pi^{2}}{6}\\
\sum_{n=1}^{\infty} \frac{1}{n^{4}} = \frac{\pi^{4}}{90}\\
\sum_{n=1}^{\infty} \frac{1}{n^{6}} = \frac{\pi^{6}}{945}
\end{align*}
etc, but there is no known analytic form for 
$$\sum_{n=1}^{\infty}\frac{1}{n^{k}}$$
when $k$ is odd.
As another rich area that would be better left for someone else elaborate on: Newton's method. For different initial values, there is still relatively very little which is known about the behavior of the Newton iterates. (Especially considering that this is one of the most well known methods for finding roots numerically.)
