The final result, in the preceeding post, can not be derived from a telescoping series [3], if is not integer (See comments at reference [1]).

This lack of generality, can be avoided, if we consider a more general definition for the Harmonic Numbers [4], extended to the complex plane, using the function:

Where is the so called digamma function, and is the Euler-Mascheroni constant.

If you take a look at the expresion (15), in the reference [2] : We can find that one asymptotic expansion for the digamma function is:

This is why the Polygonal Numbers Series sum is working:

And the polygonal numbers infinite sum, can be expressed (if ) as:

This expresion works for all , as well as for all nonreal , It also works for all , except if , and is , because is not defined for negative integers (See reference) [1]

**References:**

[1]-**Charles R Greathouse IV – **Comments @ My Math Forum Inverse Polygonal Series

[2]-**Weisstein, Eric W. “Digamma Function.” **From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/DigammaFunction.html

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

[4]-**Sondow, Jonathan and Weisstein, Eric W. “Harmonic Number.” **From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/HarmonicNumber.html

[5]-Photo **Martin Gardner, Mathematical Games, Scientific American, 211(5):126-133**, taken from http://bit-player.org/2007/hung-over
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