Abel's uniform convergence test
Encyclopedia
In classical mathematical analysis
, Abel's uniform convergence test, one of many things named after Niels Henrik Abel
, is a criterion for the uniform convergence of a series
of functions
or an improper integration
of functions
dependent on parameters. It is related to Abel's test
for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.
The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous function
s on a set E such that gn+1(x) ≤ gn(x) for all x ∈ E and positive integers n, and let {ƒn} be a sequence of real-valued functions such that the series Σƒn(x) converges uniformly on E. Then Σƒn(x)gn(x) converges uniformly on E.
Mathematical analysis
Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...
, Abel's uniform convergence test, one of many things named after Niels Henrik Abel
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
, is a criterion for the uniform convergence of a series
Series (mathematics)
A series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
or an improper integration
Improper integral
In calculus, an improper integral is the limit of a definite integral as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits....
of functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
dependent on parameters. It is related to Abel's test
Abel's test
In mathematics, Abel's test is a method of testing for the convergence of an infinite series. The test is named after mathematician Niels Abel...
for the convergence of an ordinary series of real numbers, and the proof relies on the same technique of summation by parts.
The test is as follows. Let {gn} be a uniformly bounded sequence of real-valued continuous function
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...
s on a set E such that gn+1(x) ≤ gn(x) for all x ∈ E and positive integers n, and let {ƒn} be a sequence of real-valued functions such that the series Σƒn(x) converges uniformly on E. Then Σƒn(x)gn(x) converges uniformly on E.