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Differentiability of a Series of Functions 
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#1
Nov1812, 11:51 AM

P: 22

I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (x^{n})/n^{2}, where n≥1, converges to some f(x), and that f(x) is continuous, differentiable, and integrable on [1,1].
I know how to show that f(x) is continuous, since each f_{n}(x) is continuous, and I f_{n}(x) converges uniformly. Because each f_{n}(x) is also integrable, I can also show f(x) is integrable. The trouble I'm having is proving that f(x) is differentiable. I need to show that the series of derivatives converges uniformly. However, I don't think I can use the Weierstrass MTest in this scenario. Any ideas? 


#2
Nov2012, 07:56 AM

P: 339




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