The Weierstrass Test

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The Weierstrass M – Test gives an often quick and easy method of determining whether of not a series is uniformly convergent.

Theorem (Weierstrass M – Test)

Supposeis a sequence of functions defined on a set E andis a sequence of nonnegative real numbers such thatfor allIf converges then so doesand this also converges uniformly on

Proof: ChooseSinceconverges, there is a real numbersuch that for positive integerswe haveThen for all positive integersandimplies thatfor all x in E, henceconverges uniformly on E.

Example: Show thatconverges uniformly on

so take

The last expression is a geometric series with first termand common ratio

Sincethe geometric series converges, so by comparisonconverges in this example and

converges uniformly on

Example: Show thatwhereodd andeven converges uniformly on

onso take

This is a geometric series with first termand common ratioSincethis series converges soconverges uniformly.