Fr 0568 # 64

Forum for the GRE subject test in mathematics.
rhnsrbh
Posts: 5
Joined: Sat Mar 27, 2010 5:03 am

Fr 0568 # 64

Postby rhnsrbh » Mon Mar 29, 2010 7:03 am

64. For each positive integer n, let f_n be the function defined on the interval [0,1] by f_n(x) = x^n /(1+x^n) which of the following statements is true?

I. The sequence converges pointwise.
II. The sequence converges uniformly
III. lim_n^Inf Integral_0^1 f_n(x) dx = Integral_0^1 lim_n^Inf f_n(x) dx


What is point wise and uniformity?

mrb
Posts: 25
Joined: Fri Oct 30, 2009 7:58 pm

Re: Fr 0568 # 64

Postby mrb » Mon Mar 29, 2010 11:16 am

Hi, rhnsrbh.

Roughly speaking, pointwise convergence means the functions f_n converge to a function f in the obvious way, and uniform convergence means the f_n converge to f with all points in the domain converging at "the same speed". For a precise definition see any book on basic analysis (which you should read before taking the Mathematics GRE).

In this case, the f_n converge pointwise to

f(x) = 0 if x is not 1
f(1) = 1/2

This is not continuous, and there is a theorem that then implies that this cannot be uniform convergence. Again, see any book on basic analysis.




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