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