Ramanujan Summation - Summation

Summation

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

\begin{align} {} &\frac{1}{2}f\left( 0\right) + f\left( 1\right) + \cdots + f\left( n - 1\right) + \frac{1}{2}f\left( n\right) \\ = &\frac{1}{2}\left + \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 + R_p \end{align}

Ramanujan wrote it for the case p going to infinity:

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.

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