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The product of absconverg series and bounded seq is absolutely convergent

  1. Nov 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Assume [tex]\sum_{1}^{\infty} a_n[/tex] is absolutely convergent and {bn} is bounded. Prove [tex]\sum_{1}^{\infty} a_n * b_n[/tex] is absolutely convergent


    2. Relevant equations
    A series is absolutely convergent iff the sum of | an | is convergent

    A series is convergent if for every e there is an N and P such that for all n >= N, for all p >0,
    | S_{n+p} - S{n-1} | < e. S_k is the kth partial sum, and this basically says that the sequence of partial sum must be cauchy.


    3. The attempt at a solution

    because the series of an is abosultely convergent | |a_{n+p}| - |a_{n-1}| | < e
    and bn is bounded means | b_n| < M

    [tex] \left| \left| a_{n+p} b_{n+p} \right| - \left| a_{n-1}b_{n-1} \right| \right| \leq \left| a_{n+p} b_{n+p} \right| + \left| a_{n-1} b_{n-1} \right| \leq M \left| a_{n+p} \right| + M \left| a_{n-1} \right| < \frac{Me}{2M} + \frac{Me}{2M} = e [/tex]

    done
     
  2. jcsd
  3. Nov 30, 2008 #2
    I'm confused...do you have a question?
     
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