# Prove that...(limits and series)?

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• April 23rd 2009, 05:43 AM
fardeen_gen
Prove that...(limits and series)?
Prove that:
$\lim_{n\rightarrow \infty} \sum_{i = 0}^{n}\left(\sum_{j = 0}^{i} a^j\right)^{r^i} = \lim_{n\rightarrow \infty} \prod_{j = 0}^{1}\sum_{i = 0}^{n} (a^{j}\cdot r)^i$
• April 23rd 2009, 06:58 AM
Krizalid
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.