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
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
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