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

    Solution: Let $\displaystyle S_n, T_n, R_n$ be the partial sums of the series $\displaystyle \sum_{n=1} a_n, \sum_{n=1} b_n, \sum_{n=1} c_n$ respectively. Now $\displaystyle (R_{2n-1}) = c_1 + c_2 +...+ c_{2n-1} = (a_1 +...+ a_n)+ (b_1 +...+b_{n-1}) = S_n +T_{n-1}$. Similarily, $\displaystyle (R_{2n}) = c_1 + c_2 +...+ c_{2n-1} + c_{2n} = (a_1 +...+ a_n)+ (b_1 +...+b_n) = S_n +T_n$. Since $\displaystyle \sum_{n=1} a_n$ and $\displaystyle \sum_{n=1} b_n$ converges, the sequence $\displaystyle (S_n)$ and $\displaystyle (T_n)$ converges. Since $\displaystyle (R_{2n-1})$ and $\displaystyle (R_{2n})$ converges to the same value, $\displaystyle (R_n)$ converges. Hence, the series $\displaystyle \sum_{n=1} c_n$ 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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