Sum of the Reciprocals of the Triangular Numbers

This file proves Theorem 42 from the 100 Theorems List.

We interpret “triangular numbers” as naturals of the form $\frac{k(k+1)}{2}$ for natural k. We prove that the sum of the reciprocals of the first n triangular numbers is $2 - \frac2n$.

Tags

discrete_sum

theorem Theorems100.inverse_triangle_sum (n : ℕ) : ((Finset.range n).sum fun (k : ℕ) => 2 / (↑k * (↑k + 1))) = if n = 0 then 0 else 2 - 2 / ↑n

Sum of the Reciprocals of the Triangular Numbers