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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. ... A series is convergent ... if it is convergent, to its sum.
In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. . More precisely, a real or complex series = is said to converge absolutely if = | | = for some real number .
Otherwise, any series of real numbers or complex numbers that converges but does not converge absolutely is conditionally convergent. Any conditionally convergent sum of real numbers can be rearranged to yield any other real number as a limit, or to diverge. These claims are the content of the Riemann series theorem. [31] [32] [33]
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, ... converges if and only if the sum = = converges ...
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. Here, is taken to have the value
The sum of the series is a random variable whose probability density function is close to for values between and , and decreases to near-zero for values greater than or less than . Intermediate between these ranges, at the values ± 2 {\displaystyle \pm 2} , the probability density is 1 8 − ε {\displaystyle {\tfrac {1}{8}}-\varepsilon } for ...
Absolutely convergent series are unconditionally convergent. But the Riemann series theorem states that conditionally convergent series can be rearranged to create arbitrary convergence. [4] Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The general principle is that addition of infinite sums is ...
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.