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
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