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Math Help - Series

  1. #1
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    Series

    "Suppose \sum a_n, \sum b_n are convergent. Let c_n=\sum_{k=0}^n a_kb_{n-k} and let \sum c_n converge. Prove that  (\sum_0^\infty a_n) (\sum_0^\infty b_n) =  (\sum_0^\infty c_n)."

    I can't make any headways. Any hints would be appreciated.
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  2. #2
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    Quote Originally Posted by Treadstone 71
    "Suppose \sum a_n, \sum b_n are convergent. Let c_n=\sum_{k=0}^n a_kb_{n-k} and let \sum c_n converge. Prove that  (\sum_0^\infty a_n) (\sum_0^\infty b_n) =  (\sum_0^\infty c_n)."

    I can't make any headways. Any hints would be appreciated.
    Perhaps you should look at its partial sums.
    Let _aS_n represent the nth partial sum of a_k let _bS_n represent the nth partial sum of b_k and _cS_n represent the nth partial sum of c_k.
    Thus, if we can show that the partial sums of these two series' are equal then they converge to the same number.
    Notice that:
    _aS_n=a_0+a_1+...+a_n
    _bS_b=b_0+b_1+...+b_n
    Then,
    \left(_aS_n\right)\left(_bS_n\right)=(a_0+...+a_n)  (b_0+...+b_n)=c_0+c_1+...+c_n= _cS_n
    (I might have made a mistake on the last step where I claim it is _cS_n but I do not see anything wrong)
    Q.E.D.
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  3. #3
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    The last step is definately not equal. On one side you have n^2 terms, on the other, you have 1+2+3+...+n terms.
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