Ramanujan Summation - Sum of Divergent Series

Sum of Divergent Series

Using standard extensions for known divergent series, he calculated "Ramanujan summation" of those. In particular, the sum of 1 + 2 + 3 + 4 + ⋯ is

where the notation indicates Ramanujan summation. This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation.

For even powers we have:

and for odd powers we have a relation with the Bernoulli numbers:

Those values are consistent with the Riemann zeta function.

More recently, the use of C(1) has been proposed as Ramanujan's summation, since then it can be assured that one series admits one and only one Ramanujan's summation, defined as the value in 1 of the only solution of the difference equation that verifies the condition .

This new definition of Ramanujan's summation (denoted as ) does not coincide with the earlier defined Ramanujan's summation, C(0), nor with the summation of convergent series, but it has interesting properties, such as: If R(x) tends to a finite limit when x → +1, then the series is convergent, and we have

In particular we have:

where γ is the Euler–Mascheroni constant.

Ramanujan resummation can be extended to integrals for example using the Euler-Maclaurin summation formula one can write

$\begin{align} I(n, \Lambda) &= \frac{n}{2}I(n-1, \Lambda) + \zeta(-n) - \sum_{r=1}^{\infty}\frac{B_{2r}}{(2r)!} a_{n,r}(n-2r+1) I(n-2r, \Lambda)\\ a_{n,r} &= \frac{\Gamma(n+1)}{\Gamma (n - 2r + 2)} \end{align}$

here with the application of this Ramanujan resummation lends to finite results in the renormalization of Quantum Field theories

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