Ramanujan summation

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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:

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

or simply:

$f\left( 1\right)+f\left( 2\right) +\cdots+f\left( n-2\right) + f\left( n-1\right) = C + \int_1^n f(x)\,dx + \sum_{k=1}^\infty\frac{B_{k+1}}{(k+1)!}\left(f^{(k)}(n)-f^{(k)}(0)\right)$

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

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

where the notation $(\Re)$ indicates Ramanujan summation.^[1] This formula originally appeared in one of Ramanujan's notebooks, without any notation to indicate that it was a Ramanujan summation.