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:
“He has the common feeling of his profession. He enjoys a statement twice as much if it appears in fine print, and anything that turns up in a footnote ... takes on the character of divine revelation.”
—Margaret Halsey (b. 1910)
“The force of truth that a statement imparts, then, its prominence among the hordes of recorded observations that I may optionally apply to my own life, depends, in addition to the sense that it is argumentatively defensible, on the sense that someone like me, and someone I like, whose voice is audible and who is at least notionally in the same room with me, does or can possibly hold it to be compellingly true.”
—Nicholson Baker (b. 1957)
“The new statement is always hated by the old, and, to those dwelling in the old, comes like an abyss of skepticism.”
—Ralph Waldo Emerson (1803–1882)