**TETRAHEDRAL NUMBERS SERIES:**

This post follows with the exercises on special numbers reciprocals related series, after the blog entries about Square Pyramidal Numbers and Polygonal Numbers . In fact, this example it is not very much interesting, but I wanted to write it before to deal with more difficult problems.

If we split the main fraction into others:

Solving the linear system of equations it gives:

This three series can be summed easily with the aid of the Harmonic Numbers:

If we sustitute everything in the expression for the reciprocals sum:

In the previous step we can see what does exactly means to be a “telescoping series“, the term , has vanished and there is no need to handle the Euler Mascheroni Gamma and the Digamma Function:

Then the formula for the n-th partial sum is:

And taking the limit we get:

**References:**[1]-Tetrahedral Number at- Wikipedia

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

[3] A000292-Tetrahedral (or pyramidal) numbers: C(n+2,3) = n(n+1)(n+2)/6. The On-Line Encyclopedia of Integer Sequences!

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