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February 1st, 2010, 12:19 PM  #1 
Senior Member Joined: Dec 2009 Posts: 150 Thanks: 0  Term by term differentiation
Suppose and is nondecreasing n = 1,2,3,... and that if then converges to a finite limit . Prove that S'(x) exists a.e. and that termbyterm differentiation is valid a.e. in [a,b]. The a.e. existence of S'(x) follows from the monotonicity of S(x), but I don't know how to show term by term differentiation holds a.e. I know that uniform convergence would allow this, and I know termbyterm integration holds a.e. but I don't know how to show that I can exchange the limiting operation of differentiation with the limit of the sum. Any hints/suggestions would be greatly appreciated!

February 1st, 2010, 02:53 PM  #2  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Term by term differentiation Quote:
 
February 2nd, 2010, 01:23 AM  #3 
Senior Member Joined: Jan 2009 From: Japan Posts: 192 Thanks: 0  Re: Term by term differentiation
You're asking how to prove that the derivative of a finite sum is the sum of the derivative of the terms? Start with two terms. You may or may not need to prove that the (f + g)'(x) = f'(x) + g'(x), depending on what level your course is. Then proceed by induction. 
February 2nd, 2010, 10:57 AM  #4  
Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3  Re: Term by term differentiation Quote:
I really should know how to do this...  
August 14th, 2010, 05:50 PM  #5 
Newbie Joined: Aug 2010 Posts: 2 Thanks: 0  Re: Term by term differentiation
For the term by term differentiation, you need to show that the series converges uniformly (on , where and is the set of dicontinuities of ). Try to use the wierstrass M test.


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