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Concept
Ramanujan summation
Summary
Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.
Since there are no properties of an entire sum, the Ramanujan summation functions as a property of partial sums. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:
Ramanujan wrote it for the case p going to infinity, and changing the limits of the integral and the corresponding summation:
where C is a constant specific to the series and its analytic continuation and the limits on the integral were not specified by Ramanujan, but presumably they were as given above. Comparing both formulae and assuming that R tends to 0 as x tends to infinity, we see that, in a general case, for functions f(x) with no divergence at x = 0:
where Ramanujan assumed By taking we normally recover the usual summation for convergent series. For functions f(x) with no divergence at x = 1, we obtain:
C(0) was then proposed to use as the sum of the divergent sequence. It is like a bridge between summation and integration.
The most common application of Ramanujan summation is for the Riemann zeta function ζ(z), in which the Ramanujan summation of the function has the same value as ζ(s) for all the values of , even for those for which the first function is divergent, which is equivalent to doing analytic continuation or, alternatively, applying smoothed sums.
The convergent version of summation for functions with appropriate growth condition is then:
In the following text, indicates "Ramanujan summation". This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it exemplified a novel method of summation.
For example, the of 1 − 1 + 1 − ⋯ is:
Ramanujan had calculated "sums" of known divergent series.
Official source
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