Statement
Suppose that is a sequence of real- or complex-valued functions defined on a set, and that there is a sequence of positive numbers satisfying
for all and all . Suppose also that the series
is convergent.
Then the series
converges uniformly on .
Remark The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the above conditions, the set is a topological space and the functions are continuous on, then the series converges to a continuous function.
Read more about this topic: Weierstrass M-test
Famous quotes containing the word statement:
“Truth is that concordance of an abstract statement with the ideal limit towards which endless investigation would tend to bring scientific belief, which concordance the abstract statement may possess by virtue of the confession of its inaccuracy and one-sidedness, and this confession is an essential ingredient of truth.”
—Charles Sanders Peirce (1839–1914)
“It is commonplace that a problem stated is well on its way to solution, for statement of the nature of a problem signifies that the underlying quality is being transformed into determinate distinctions of terms and relations or has become an object of articulate thought.”
—John Dewey (1859–1952)
“Truth is used to vitalize a statement rather than devitalize it. Truth implies more than a simple statement of fact. “I don’t have any whisky,” may be a fact but it is not a truth.”
—William Burroughs (b. 1914)