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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.
An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
Similarly, in a series, any finite groupings of terms of the series will not change the limit of the partial sums of the series and thus will not change the sum of the series. However, if an infinite number of groupings is performed in an infinite series, then the partial sums of the grouped series may have a different limit than the original ...
Each series expansion of the integrand contributes one sum. [15] The summation indices (variables) are the indices that index terms in a series expansion. In the example, there are 3 summation indices , and because the integrand is a product of 3 series expansions. [16]
In mathematics, the Euler–Maclaurin formula is a formula for the difference between an integral and a closely related sum.It can be used to approximate integrals by finite sums, or conversely to evaluate finite sums and infinite series using integrals and the machinery of calculus.
For example, the full zeta function exists at = (and is therefore finite there), but the corresponding series would be + + + …, whose partial sums would grow indefinitely large. The zeta function values listed below include function values at the negative even numbers ( s = −2 , −4 , etc. ), for which ζ ( s ) = 0 and which make up the so ...
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators.
[3] [4] The boundary conditions are usually imposed by the Simultaneous-Approximation-Term (SAT) technique. [5] The combination of SBP-SAT is a powerful framework for boundary treatment. The method is preferred for well-proven stability for long-time simulation, and high order of accuracy.