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Math Help - Continuity of a Sum on R

  1. #1
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    Continuity of a Sum on R

    Show SUM( (sin(nx)/n^2) * x^3) is continuous on R
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  2. #2
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    Quote Originally Posted by cp05 View Post
    Show SUM( (sin(nx)/n^2) * x^3) is continuous on R


    This is a series of functions which converges everywhere (Weierstrass M Test) times the function x^3 , thus...

    Tonio
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  3. #3
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    why does the fact that it converges everywhere...namely, it's uniformly convergent, show that (sin nx)/n^2 is continuous?
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    Quote Originally Posted by cp05 View Post
    why does the fact that it converges everywhere...namely, it's uniformly convergent, show that (sin nx)/n^2 is continuous?

    Weierstrass M-test - Wikipedia, the free encyclopedia

    Read with particular care the 9th line.

    Tonio
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  5. #5
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    I read the wikipedia line....but I don't see why it's continuous if it converges uniformly
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  6. #6
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    Quote Originally Posted by cp05 View Post
    I read the wikipedia line....but I don't see why it's continuous if it converges uniformly

    Because the limit function of continuous uniformly convergent functions is continuous...!! If you want you can prove this by yourself: it's pretty easy, or else you can google or check any decent analysis book. It's pretty standard stuff.

    Tonio
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