Posted by: Gary Ernest Davis on: November 5, 2013

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We learn early in a study of infinite series that the geometric series sums to 1.

Sometimes you will see this sort of reasoning:

so so

which is somewhat suspect in light of Euler’s “argument”:

soÂ .

We need first to know thatÂ converges

the term ofÂ divided by the term is 1/2 <1.

Another application of the ratio test shows that theÂ series is absolutely convergent:

theÂ of this series is , the term isÂ and their ratio isÂ which approaches as .

If we denoteÂ then a simple, and legitimate, calculation shows that which we Â know to be .

In fact, for any natural number the ratio test shows that the series is absolutely convergent.

A MathematicaÂ® calculation:

**TableForm[Table[{p, Limit[Sum[k^p/2^k, {k, 1, n}], n -> Infinity]}, {p, 0, 10}], TableHeadings -> {None, {“p”, “S(p)”}}]**

Â yields the following results for :

The first curious thing is that the results are all whole numbers. Why is that?

The second curious thing is that if we enter this sequence of whole numbers into theÂ The On-Line Encyclopedia of Integer SequencesÂ we get a match:

these whole numbers match the number of necklaces of partitions of p+1 labeled beads. They also match the sequence of cumulants of the probability distribution of the number of tails before the first head in a sequence of fair coin tosses. Is that right? If so, why?Â is the first really interesting case.

A detailed account of these and similar series is dealt with byÂ MirceaÂ CÃ®rnu “DeterminantalÂ formulas forÂ sumÂ of generalizedÂ arithmetic-geometric series“Â Boletn de la Asociacion Matematica Venezolana, Vol. XVIII, No. 1 (2011), 13-25.

Enjoy! It’s connections like these that give mathematicians a buzz.

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