1 + 2 + 3 + … = – 1/12 and Related Results

The seemingly bizarre result in the title of this article has surprisingly found its way into some branches of advanced physics including string theory.  Its ‘proof ‘ is demonstrated in one of the Numberphile videos on YouTube where divergent series are manipulated in an easy-going manner, but the series can also be regarded in a more mathematically acceptable light as an expression of the Riemann zeta function ζ(z) evaluated at z = 1.

It occurred to me that it should also be possible to sum powers of the natural numbers using the same carefree operations on divergent series as in the Numberphile video.  In a discussion with the title Some Fun with Divergent Series, I first evaluate the sum of the squares of the natural numbers S2 and then, after a digression on Abel summation and the Riemann zeta function, I generalise the method to find the sum Sn of the series of natural numbers raised to the nth power.  Initially a recursion relation is found for a related quantity Tn  , the sum of powers of the natural numbers with alternating signs.  This leads directly to the formula for Sn and its connection with the well-known Bernoulli numbers.

After completing this article I discovered a WordPress blog by the eminent Australian/American mathematician Terry Tao. In one of his published articles there, entitled The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, he explores the summation of powers of the natural numbers in much greater detail and, of course, with considerably more insight than anything I could offer.  Nevertheless I have kept my approach posted here as it is a natural generalisation of the informal method shown in the Numberfile video referred to above.

Since I have been unable to find a way to embed a PDF file in a WordPress blog, the best I can do is to provide this link to Some Fun with Divergent Series.


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