Is there a π solution for ∑i^{-i+1}

The sum∑i^{-i+1} converges very fast as compared with∑i^{-i}, but its value is very close to Euler's solution for the Basel Problem. This begs the question, can there be a solution involving π or other identities ?

Another Proof of Basel Problem

In this paper, we present a new solution to the Basel problem using a complex line integral of log(1+z)/z.

Basel Problem

Summary A rigorous elementary proof of the Basel problem [6, 1] $$\sum\nolimits_{n = 1}^\infty {{1 \over {n^2 }} = {{\pi ^2 } \over 6}} $$ is formalized in the Mizar system [3]. This theorem is item #14 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

Basel Problem – Preliminaries

Summary In the article we formalize in the Mizar system [4] preliminary facts needed to prove the Basel problem [7, 1]. Facts that are independent from the notion of structure are included here.