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Math Help - Series of Functions

  1. #1
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    Series of Functions

    (1)

    (a) Show that g(x)= \sum^{\infty}_{n=1}{cos\frac{(2^nx)}{2^n}} is continuous on all of \mathbb{R}

    (b) Prove that h(x)= \sum^{\infty}_{n=1}{\frac{x^n}{n^2}} is continuous on [-1,1].

    (2) Observe that the series f(x)=\sum^{\infty}_{n=1}{\frac{x^n}{n}}=x+\frac{x^  2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+... converges for every x in the half-open interval [0,1) but does not converge when x=1. For a fixed x_0 \in (0,1), explain how we can still use the Weierstrass M-Test to prove that f is continuous at x_0.

    (3) Let h(x)=\sum^{\infty}_{n=1}{\frac{1}{x^2+n^2}}

    Show that h is a continuous function defined on all of \mathbb{R}.

    Thanks!
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  2. #2
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    You prove these problems uses the Weierstrass test. For example, in 1(a) we see that \left| \tfrac{\cos 2^n x}{2^n} \right| \leq \tfrac{1}{2^n} and the series of \tfrac{1}{2^n} converges. Thus, this series of fuctions converges uniformly. Since each function is continous we see that the limit (the infinite sum) is continous because the uniform limit of continous functions is continous.
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