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Generating function – Formal power series; coefficients encode information about a sequence indexed by natural numbers; Perron's formula – Formula to calculate the sum of an arithmetic function in analytic number theory; Renormalization – Method in physics used to deal with infinities; 1 + 1 + 1 + 1 + ⋯ – Divergent series
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
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.
In the mathematics of convergent and divergent series, Euler summation is a summation method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series Σa n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original ...
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.
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.
In mathematics, summation by parts transforms the summation of products of sequences into other summations, often simplifying the computation or (especially) estimation of certain types of sums. It is also called Abel's lemma or Abel transformation, named after Niels Henrik Abel who introduced it in 1826. [1]
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