|Nov18-12, 11:51 AM||#1|
Differentiability of a Series of Functions
I'm working on a problem where I need to show that the series of functions, f(x) = Ʃ (xn)/n2, 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 fn(x) is continuous, and I fn(x) converges uniformly. Because each fn(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 M-Test in this scenario. Any ideas?
|Nov20-12, 07:56 AM||#2|
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