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Math Help - Convergence

  1. #1
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    Convergence

    Suppose that a_k \to \infty as k \to \infty. Prove that \displaystyle\sum_{k=1}^{\infty}  a_k converges if and only if the series \displaystyle\sum_{k=1}^{\infty} (a_{2k} + a_{2k+1}) converges.
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  2. #2
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    You realize its the same series sans the first term right? So \sum_{k=1}^N a_k = a_1 + \sum_{k=1}^{(N-1)/2}(a_{2k} + a_{2k+1}) if N is odd and \sum_{k=1}^N a_k + a_{N+1} = a_1 + \sum_{k=1}^{N/2}(a_{2k} + a_{2k+1})
    if N is even...

    Also, if the terms of the sum are going to infinity, doesn't the series diverge?
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  3. #3
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    Quote Originally Posted by maddas View Post
    \sum_{k=1}^N a_k = a_1 + \sum_{k=1}^{(N-1)/2}(a_{2k} + a_{2k+1}) if N is odd and \sum_{k=1}^N a_k + a_{N+1} = a_1 + \sum_{k=1}^{N/2}(a_{2k} + a_{2k+1})
    if N is even...
    I definitely do not understand this. Could anyone explain?
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  4. #4
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    Quote Originally Posted by failureequalslearn View Post
    i definitely do not understand this. Could anyone explain?
    \sum_{j=1}^{10}j1+2+3+4+5+6+7+8+9+10=1+(2+3)+(3+4)  +(5+6)+(7+8)+(9+10) =1+\sum_{j=1}^{\frac{10}{2}}(2j+2j+1)
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  5. #5
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    Quote Originally Posted by FailureEqualsLearn View Post
    Suppose that a_k \to \infty as k \to \infty. Prove that \displaystyle\sum_{k=1}^{\infty}  a_k converges if and only if the series \displaystyle\sum_{k=1}^{\infty} (a_{2k} + a_{2k+1}) converges.
    Do you all realize that if \lim _{k \to \infty } (a_k ) = \infty then \sum\limits_k {a_k } must diverge.
    It cannot converge.
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  6. #6
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    Wow I screwed this thread up from the get go. It should read a_k \to 0 as k \to \infty. My apologies =[.
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