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Math Help - Convergence of a Series (proof)

  1. #1
    Senior Member roninpro's Avatar
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    Convergence of a Series (proof)

    Hello. I am having some difficulty proving the following:

    If \sum_{n=1}^\infty a_n converges and \{b_n\} is monotonic and bounded, then \sum_{n=1}^\infty a_nb_n converges.

    I tried using the Cauchy Criterion and partial summation to handle this, but I haven't had any luck. Your insight would be appreciated.
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  2. #2
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    Quote Originally Posted by roninpro View Post
    Hello. I am having some difficulty proving the following:

    If \sum_{n=1}^\infty a_n converges and \{b_n\} is monotonic and bounded, then \sum_{n=1}^\infty a_nb_n converges.

    I tried using the Cauchy Criterion and partial summation to handle this, but I haven't had any luck. Your insight would be appreciated.

    It's a result of Dirichlet's test: if \sum\limits_{n=1}^\infty a_n is a bounded series (i.e., its partials sums sequence is bounded) and if \{b_n\}_{n=1}^\infty is a descending monotone sequence that converges to zero, then \sum\limits_{n=1}^\infty a_nb_n converges:

    Since \{b_n\} is monotone and bounded it converges to a finite limit, say L. Assume it is monotone ascending (if it is descending it is very simmlilar) , so we get (L-b_n) \xrightarrow [n\to \infty] {} 0 monotonically descending and \sum\limits_{n=1}^\infty a_n is bounded because it is convergent, and thus by Dirichlet's test the series \sum\limits_{n=1}^\infty a_n(L-b_n) converges, but then:

    \sum\limits_{n=1}^\infty a_nL-\sum\limits_{n=1}^\infty a_n(L-b_n) is the difference of two convergent series and thus it converges, and \sum\limits_{n=1}^\infty a_nL-\sum\limits_{n=1}^\infty a_n(L-b_n)=\sum\limits_{n=1}^\infty a_nb_n by arithmetic of limts, and we're done.

    Tonio
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