The Weierstrass M - Test

Weierstrass's M – test provides a way to prove uniform convergence for the sum of a sequence.

Letbe a sequence of functions on a setand suppose that there is a sequence of positive termssuch that

  1. forand all

  2. is convergent

    Then the seriesis uniformly convergent on

    Proof: Note that by assumption 1 and 2,is convergent for eachby the comparison test. henceis convergent for eachby the absolute convergence test and so the sum functionexists for eachWe note the partial sum functions by

    Then for

    by the triangle inequality

    by assumption 1

    By assumption 2,asso uniform convergence follows..

    Example: Prove thatconverges uniformly onfor


    so that assumption 1 above is satisfied with


    so that assumption 2 is satisfied and the sum converges uniformly on