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In mathematics, a series is the sum of the terms of an infinite sequence of numbers. More precisely, an infinite sequence (,,, …) defines a series S that is denoted = + + + = =. The n th partial sum S n is the sum of the first n terms of the sequence; that is,
In summary, series addition and scalar multiplication gives the set of convergent series and the set of series of real numbers the structure of a real vector space. Similarly, one gets complex vector spaces for series and convergent series of complex numbers. All these vector spaces are infinite dimensional.
While most of the tests deal with the convergence of infinite series, they can also be used to show the convergence or divergence of infinite products. This can be achieved using following theorem: Let { a n } n = 1 ∞ {\displaystyle \left\{a_{n}\right\}_{n=1}^{\infty }} be a sequence of positive numbers.
In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, and rearranged such that the new series diverges.
The sequence of partial sums obtained by grouping is a subsequence of the partial sums of the original series. The convergence of each absolutely convergent series is an equivalent condition for a normed vector space to be Banach (i.e.: complete).
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.
Agnew's theorem describes rearrangements that preserve convergence for all convergent series. The Lévy–Steinitz theorem identifies the set of values to which a series of terms in R n can converge. A typical conditionally convergent integral is that on the non-negative real axis of (see Fresnel integral).
If a series is convergent but not absolutely convergent, it is called conditionally convergent. An example of a conditionally convergent series is the alternating harmonic series. Many standard tests for divergence and convergence, most notably including the ratio test and the root test, demonstrate absolute convergence.