This post continues with the work on some special, sets of integers, related series:

The square pyramidal numbers expression can be found on [1] and it is:

The Square pyramidal numbers reciprocal finite serie is:

Splitting the main fraction into others:

Solving the equations gives:

Substituing the series (ii), and the expression for the Harmonic Numbers (i):

Taking into account the expressions: (i),(iv) and (vi), for the digamma function:

Calculating the limit, and using (vii):

(see reference [4])

**NOTES:**
*(i) Harmonic Numbers and Digamma Function at integer values:*

*(ii) Changing series into Harmonic Numbers:*

*(iii) Changing one series into another:*

*(iv) Digamma function at half-integer values:*

*(v) A Digamma function property:*

*(vi) Another expression for Digamma function at half-integer values:*

Using *(iii)*,* (iv)* and *(v)*:

*(vii) Limit for Digamma function at half-integer values:*

The last serie limit can be derived from the Mercator-Mengoli infinite series for . See [3].

This proof is interesting enough for another entry on the blog.

So:

**References:**
[1]-Square Pyramidal Numbers @ Wikipedia Square Pyramidal Numbers

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

[3]-Collection of formulae for log 2-Numbers Constants and Computation-Xavier Gourdon and Pascal Sebah.

[4]-A159354-Decimal expansion of 18-24*log(2) The On-Line Encyclopedia of Integer Sequences!

[5]-A000330-Square pyramidal numbers The On-Line Encyclopedia of Integer Sequences!

**Archives:**
[a]-041109-SQUARE PYRAMIDAL NUMBERS RECIPROCALS SUM.nb