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.
Summation 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:
1 2 f ( 0 ) + f ( 1 ) + ⋯ + f ( n − 1 ) + 1 2 f ( n ) = f ( 0 ) + f ( n ) 2 + ∑ k = 1 n − 1 f ( k ) = ∑ k = 0 n f ( k ) − f ( 0 ) + f ( n ) 2 = ∫ 0 n f ( x ) d x + ∑ k = 1 p B 2 k ( 2 k ) ! [ f ( 2 k − 1 ) ( n ) − f ( 2 k − 1 ) ( 0 ) ] + R p {\displaystyle {\begin{aligned}{\frac {1}{2}}f(0)+f(1)+\cdots +f(n-1)+{\frac {1}{2}}f(n)&={\frac {f(0)+f(n)}{2}}+\sum _{k=1}^{n-1}f(k)=\sum _{k=0}^{n}f(k)-{\frac {f(0)+f(n)}{2}}\\&=\int _{0}^{n}f(x)\,dx+\sum _{k=1}^{p}{\frac {B_{2k}}{(2k)!}}\left[f^{(2k-1)}(n)-f^{(2k-1)}(0)\right]+R_{p}\end{aligned}}}
Ramanujan wrote this again for different limits of the integral and the corresponding summation for the case in which p goes to infinity:
∑ k = a x f ( k ) = C + ∫ a x f ( t ) d t + 1 2 f ( x ) + ∑ k = 1 ∞ B 2 k ( 2 k ) ! f ( 2 k − 1 ) ( x ) {\displaystyle \sum _{k=a}^{x}f(k)=C+\int _{a}^{x}f(t)dt+{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)}
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