let (gn) be a sequence of functions on D such that gn+1(x)< gn(x) for all x in D. if (gn)-> 0 uniformly on D, show that Σ (-1)^n(gn) converges uniformly on D

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- Apr 29th 2010, 02:10 PMlavender87sequences and series of functions
let (gn) be a sequence of functions on D such that gn+1(x)< gn(x) for all x in D. if (gn)-> 0 uniformly on D, show that Σ (-1)^n(gn) converges uniformly on D

- Apr 30th 2010, 07:33 AMCaptainBlack
The series is pointwise convergence by the alternating series test (since the series is decreasing all the terms are non-negative on etc).

The conditions imply that there exists a decreasing positive sequence as such that for all .

Then (since the remainder for a truncated alternating series is bounded by the absolute value of the first neglected term):

etc.

CB