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Math Help - Prove that...(limits and series)?

  1. #1
    Super Member fardeen_gen's Avatar
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    Prove that...(limits and series)?

    Prove that:
    \lim_{n\rightarrow \infty} \sum_{i = 0}^{n}\left(\sum_{j = 0}^{i} a^j\right)^{r^i} = \lim_{n\rightarrow \infty} \prod_{j = 0}^{1}\sum_{i = 0}^{n} (a^{j}\cdot r)^i
    Last edited by fardeen_gen; May 22nd 2009 at 10:14 PM.
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    to reverse summation order we only play with inqualities. (this trick is also useful when reversing integration order on double integrals.)

    thus, for both sums, we have 0\le i\le n & 0\le j\le i, now just combine these inequalities to get 0\le j\le i\le n, and finally split this multiple inequality into two ones to get the desired order of summation.
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