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Limit of a convergent series and a divergent sequence

  1. Feb 25, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that if:


    [tex]\sum_{k=1}^\infty a_k[/tex] converges and,

    [tex]\sum_{k=1}^\infty a_k b_k[/tex] converges, then

    [tex]lim_{m\to\infty} b_m \sum_{k=m}^\infty a_k = 0[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I only have an idea why this is true--[tex]\sum a_k[/tex] converges, so the tails become small, and [tex]\sum a_k b_k[/tex] converges, so [tex]a_k[/tex] shrinking dominates [tex]b_k[/tex] blowing up. I know that [tex]|b_k| < \epsilon / |a_k|[/tex] for any epsilon and large k, and I know that [tex]\left| \sum_{m=k}^\infty a_m \right| < \epsilon[/tex] for large k, but I can't find the way to combine these two statements.
  2. jcsd
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