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Math Help - regarding to Weierstrass M-test

  1. #1
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    regarding to Weierstrass M-test

    If each fn is continuous on [a, b] and (sum of)fn converges uniformly on [a, b] then (sum of)Mn
    converges, where Mn = max|fn(x)|for x belongs to [a,b].

    Is it true?

    Thx in advance

    .
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  2. #2
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    Quote Originally Posted by wheatkc View Post
    If each fn is continuous on [a, b] and (sum of)fn converges uniformly on [a, b] then (sum of)Mn
    converges, where Mn = max|fn(x)|for x belongs to [a,b].

    Is it true?
    No, it's not true. But to find a counterexample you have to use some test other than the M-test to prove the uniform convergence. That probably means using the uniform version of the Dirichlet or Abel tests for convergence. A typical counterexample is the series \sum\frac{(-x)^n}n on the interval [0,1]. This converges uniformly by the uniform Dirichlet test. But \max\{|(-x)^n/n|:x\in[0,1]\} = 1/n and \sum 1/n does not converge.
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    thx...
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