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

source

theorem Theorem100.inverse_triangle_sum (n : ℕ) :

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

Sum of the Reciprocals of the Triangular Numbers