# What mathematical expressions took a long time for a closed form to be found?

The Basel problem, $$\sum_{n=1}^\infty{\frac{1}{n^2}}$$, took 90 years for a closed form, $$\frac{\pi^2}{6}$$, to be found.

I'm curious to know what other mathematical expressions, especially those involving limits, took a long time for a closed form to be found. (By "found", I mean discovered and verified.)

• Cardano's formula for the roots of a cubic. It is hard to say when the idea of a closed form algebraic solution was first conceived, but Omar Khayyam already talked of its desirability c. 1100. The formula was only derived c. 1540. Jun 19 at 8:55
• In more recent times, one of Ramanujan's closed form expressions for a series from his lost notebook (written in 1919-20) was only proved in 2019 by Berndt, Li, and Zaharescu, see (1.10). In fairness, the notebook was only found in 1976. Jun 19 at 9:35

The theorem is an identity for formal series: $$\prod_{n=1}^\infty(1-x^n)=\sum_{-\infty}^\infty(-1)^nx^{n(3n-1)/2}.$$