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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.
The Cauchy convergence test is a method used to test infinite series for convergence. It relies on bounding sums of terms in the series. It relies on bounding sums of terms in the series. This convergence criterion is named after Augustin-Louis Cauchy who published it in his textbook Cours d'Analyse 1821.
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.
For example, many asymptotic expansions are derived from the formula, and Faulhaber's formula for the sum of powers is an immediate consequence. The formula was discovered independently by Leonhard Euler and Colin Maclaurin around 1735. Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.
In modern mathematics, the sum of an infinite series is defined to be the limit of the sequence of its partial sums, if it exists. The sequence of partial sums of Grandi's series is 1, 0, 1, 0, ..., which clearly does not approach any number (although it does have two accumulation points at 0 and 1). Therefore, Grandi's series is divergent
In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 ≤ a i , 1 ≤ a i , 2 ≤ ⋯ {\displaystyle 0\leq a_{i,1}\leq a_{i,2}\leq \cdots } , it says that taking the sum and the supremum can be interchanged.
There are two slightly different versions of Abel's test – one is used with series of real numbers, and the other is used with power series in complex analysis. Abel's uniform convergence test is a criterion for the uniform convergence of a series of functions dependent on parameters.
The technique of the previous example may also be applied to other Dirichlet series. If a n = μ ( n ) {\displaystyle a_{n}=\mu (n)} is the Möbius function and ϕ ( x ) = x − s {\displaystyle \phi (x)=x^{-s}} , then A ( x ) = M ( x ) = ∑ n ≤ x μ ( n ) {\displaystyle A(x)=M(x)=\sum _{n\leq x}\mu (n)} is Mertens function and