Let there be the n-th Polygonal Number [4] of s sides, then:

For s=3, we get the formula for Triangular Numbers

and with s=4 then we get the Squares:

Polygonal numbers hold for the next identity:

The sum of the inverse of the first k polygonal numbers with side s, is:

And its infinite series sum:

Than this infinite series is convergent, can be proved, using some convergence test:

Raabe´s convergence Test:

A series with a lower number of sides, upper bounds, series with higher sides: So the convergence in triangular numbers, implies the convergence of the remaining polygonal numbers series:

If

The inverse series with Triangular numbers is a telescoping series [3]:

The squares sum is the so called Basel Problem, [1] [2], related with the Riemann Zeta Function:

If , then:

The series is a more general case of a telescoping series [3], related with the Harmonic Numbers.

Example: In the hexagonal numbers case:

**Archives:**
[a]-031009-INVERSE POLYGONAL NUMBERS SERIES.nb

**References:**
[1]-**Estimating Basel Problem**@ MAA Online How Euler did it, Ed.Sandifer

[2]-**Basel Problem** @ Wikipedia Basel Problem

[3]-**Telescoping Series** @ Wikipedia Telescoping Series

[4]-**Polygonal Numbers** @ Wikipedia Polygonal Number

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