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Math Help - Convergent Series and Partial Sums

  1. #1
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    Convergent Series and Partial Sums

    I have a solution to the following problem but I am not sure whether it is correct or not. Can someone help me check my solution. Thanks! (The series in this question are all infinite series)

    Question: Let \sum_{n=1} a_n and \sum_{n=1} b_n be convergent series. For each n \in \mathbb{N}, let c_{2n-1} = a_n and c_{2n} = b_n. Prove that \sum_{n=1} c_n converges.

    Solution: Let S_n, T_n, R_n be the partial sums of the series \sum_{n=1} a_n, \sum_{n=1} b_n, \sum_{n=1} c_n respectively. Now (R_{2n-1}) = c_1 + c_2 +...+ c_{2n-1} = (a_1 +...+ a_n)+ (b_1 +...+b_{n-1}) = S_n +T_{n-1}. Similarily, (R_{2n}) = c_1 + c_2 +...+ c_{2n-1} + c_{2n} = (a_1 +...+ a_n)+ (b_1 +...+b_n) = S_n +T_n. Since \sum_{n=1} a_n and \sum_{n=1} b_n converges, the sequence (S_n) and (T_n) converges. Since (R_{2n-1}) and (R_{2n}) converges to the same value, (R_n) converges. Hence, the series \sum_{n=1} c_n converges.
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  2. #2
    Super Member girdav's Avatar
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    Re: Convergent Series and Partial Sums

    Yes, it's correct.
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