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e. In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. [ 1 ] The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures in combinatorics through generating functions.
List of mathematical series. This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. is a Bernoulli polynomial. is an Euler number. is the Riemann zeta function. is the gamma function. is a polygamma function. is a polylogarithm.
In mathematics, a telescoping series is a series whose general term is of the form , i.e. the difference of two consecutive terms of a sequence . [ 1 ] As a consequence the partial sums only consists of two terms of after cancellation. [ 2 ][ 3 ] The cancellation technique, with part of each term cancelling with part of the next term, is known ...
Calculus. The fundamental theorem of calculus is a theorem that links the concept of differentiating a function (calculating its slopes, or rate of change at each point in time) with the concept of integrating a function (calculating the area under its graph, or the cumulative effect of small contributions). Roughly speaking, the two operations ...
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence of real numbers converges to ...
The root test was developed first by Augustin-Louis Cauchywho published it in his textbook Cours d'analyse(1821).[1] Thus, it is sometimes known as the Cauchy root testor Cauchy's radical test. For a series. ∑n=1∞an{\displaystyle \sum _{n=1}^{\infty }a_{n}} the root test uses the number.
t. e. In mathematics, the integral test for convergence is a method used to test infinite series of monotonic terms for convergence. It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
Table of Newtonian series. In mathematics, a Newtonian series, named after Isaac Newton, is a sum over a sequence written in the form. where. is the binomial coefficient and is the falling factorial. Newtonian series often appear in relations of the form seen in umbral calculus.
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