# Ramanujan summation

Ramanujan summation is a technique invented by the mathematician Srinivasa Ramanujan for assigning a sum to infinite divergent series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties which make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined.

Ramanujan summation essentially is a property of the partial sums, rather than a property of the entire sum, as it doesn't exist. If we take the Euler–Maclaurin summation formula together with the correction rule using Bernoulli numbers, we see that:

:$frac\left\{fleft\left( 0 ight\right) \right\}\left\{2\right\}+fleft\left( 1 ight\right) +cdots+fleft\left( n-1 ight\right) +frac\left\{fleft\left( n ight\right) \right\}\left\{2\right\} =frac\left\{fleft\left( 0 ight\right) +fleft\left( n ight\right) \right\}\left\{2\right\}+sum_\left\{k=1\right\}^\left\{n-1\right\}fleft\left(k ight\right) = int_0^n f\left(x\right),dx + sum_\left\{k=1\right\}^pfrac\left\{B_\left\{k+1\left\{\left(k+1\right)!\right\}left\left(f^\left\{\left(k\right)\right\}\left(n\right)-f^\left\{\left(k\right)\right\}\left(0\right) ight\right)+R$

or simply::$fleft\left( 1 ight\right)+fleft\left( 2 ight\right) +cdots+fleft\left( n-2 ight\right) + fleft\left( n-1 ight\right)= C + int_1^n f\left(x\right),dx + sum_\left\{k=1\right\}^inftyfrac\left\{B_\left\{k+1\left\{\left(k+1\right)!\right\}left\left(f^\left\{\left(k\right)\right\}\left(n\right)-f^\left\{\left(k\right)\right\}\left(0\right) ight\right)$

Where C is a constant specific to the series and its analytic continuum. This he proposes to use as the sum of the divergent sequence. It is like a bridge between summation and integration. Using standard extensions for known divergent series, he calculated "Ramanujan summation" of those. In particular, the sum of 1 + 2 + 3 + 4 + · · · is

:$1+2+3+cdots = -frac\left\{1\right\}\left\{12\right\} \left(Re\right)$

where the notation $\left(Re\right)$ indicates Ramanujan summation. [ Éric Delabaere, [http://algo.inria.fr/seminars/sem01-02/delabaere2.pdf Ramanujan's Summation] , "Algorithms Seminar 2001–2002", F. Chyzak (ed.), INRIA, (2003), pp. 83–88.] 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:

:$1+2^\left\{2k\right\}+3^\left\{2k\right\}+cdots = 0 \left(Re\right)$

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

:$1+2^\left\{2k-1\right\}+3^\left\{2k-1\right\}+cdots = -frac\left\{B_\left\{2k\left\{2k\right\} \left(Re\right).$

* Borel summation
* Cesàro summation
* Ramanujan's sum

References

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