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
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