Ramanujan's formulas for $1/\pi$ and their generalizations remain an amazing topic, with many mathematical challenges.
Recently it was observed that the formulas possess spectacular `supercongruence' counterparts.
For example, truncating the sum in Ramanujan's formula
to the first $p$ terms correspond to the congruence
valid for any prime $p>3$.
Some supercongruences were shown earlier through a tricky use of classical hypergeometric identities or the Wilf-Zeilberger method of creative telescoping.
The particular example displayed above (and many other entries) were resistant to such techniques.
In joint work with Victor Guo we develop a new method of `creative microscoping' that simultaneously proves both the underlying Ramanujan's formula and its finite supercongruence counterparts.
The main ingredient is an asymptotic analysis of $q$-analogues of Ramanujan's formulas at roots of unity.