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For a power series f defined as: = = (),where a is a complex constant, the center of the disk of convergence,; c n is the n-th complex coefficient, and; z is a complex variable.; The radius of convergence r is a nonnegative real number or such that the series converges if
In mathematics, convergence tests are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series =. List of tests
The Riemann series theorem states that if a series converges conditionally, it is possible to rearrange the terms of the series in such a way that the series converges to any value, or even diverges. Agnew's theorem characterizes rearrangements that preserve convergence for all series.
Consider f an integrable function on the interval ... Charles Louis Fefferman, "Pointwise convergence of Fourier series", Ann. of Math. 98 (1973), 551–571.
A stronger notion of convergence of a series of functions is uniform convergence. A series converges uniformly in a set if it converges pointwise to the function at every point of and the supremum of these pointwise errors in approximating the limit by the th partial sum,
Note that sometimes a series like this is called a power series "around p", because the radius of convergence is the radius R of the largest interval or disc centred at p such that the series will converge for all points z strictly in the interior (convergence on the boundary of the interval or disc generally has to be checked separately).
Convergence in distribution is the weakest form of convergence typically discussed, since it is implied by all other types of convergence mentioned in this article. However, convergence in distribution is very frequently used in practice; most often it arises from application of the central limit theorem .
In a topological abelian group, convergence of a series is defined as convergence of the sequence of partial sums. An important concept when considering series is unconditional convergence, which guarantees that the limit of the series is invariant under permutations of the summands.