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anonymous
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 ***Spoiler*** inequality   « on: Jun 23rd, 2003, 4:11pm » Quote Modify Remove

$\begin{split} \sum_{n=1}^{\infty} (a_n-a_{n+1})a_{n+2} \\ = \sum_{n=1}^{\infty} (a_1^{2^{n-1}}-a_1^{2^n})a_1^{2^{n+1}} \\ \leq \sum_{i=1}^{\infty} (a_1^i-a_1^{i+1})a_1^{2i+2} \\ = (1-a_1)a_1^2\sum_{i=1}^{\infty} a_1^{3i} \\ = (1-a_1)a_1^5/(1-a_1^3) \\ = a_1^5/(1+a_1+a_1^2) \\ < 1/3 \end{split}$

The second to third line uses telescopic sum, and there is a geometric series in the fourth line.
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wowbagger
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 Re: ***Spoiler*** inequality   « Reply #1 on: Jun 24th, 2003, 5:17am » Quote Modify

For those who aren't used to reading LaTeX:

 « Last Edit: Jun 24th, 2003, 5:17am by wowbagger » IP Logged

James Fingas
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 Re: ***Spoiler*** inequality   « Reply #2 on: Jun 24th, 2003, 9:26am » Quote Modify

How do we justify the third line? It's not true that every term in the new sum is larger than the corresponding term in the old sum. For instance, with i=n=2 and a1=0.99, the terms are 0.017997 for the old and 0.009415 for the new.

I am assuming that you have a good justification ... maybe I don't understand telescopic sums like I thought I did?
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anonymous
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 Re: ***Spoiler*** inequality   « Reply #3 on: Jun 24th, 2003, 10:27am » Quote Modify Remove

I'll take n=3 as an example

when n=3,
$\begin{split} (a^{2^{n-1}}-a^{2^n})a^{2^{n+1}} \\ =(a^4-a^8)a^{16} \\ =(a^4-a^5)a^{16}+(a^5-a^6)a^{16}+(a^6-a^7)a^{16}+(a^7-a^8)a^{16} \\ <(a^4-a^5)a^{10}+(a^5-a^6)a^{12}+(a^6-a^7)a^{14}+(a^7-a^8)a^{16} \\ =\sum_{i=2^{n-1}}^{2^n-1}(a^i-a^{i+1})a^{2i+2} \end{split}$

When n is summed from n=1 to infinity, then i is also summed from i=1 to infinity.
 « Last Edit: Sep 7th, 2003, 2:12pm by Icarus » IP Logged
James Fingas
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 Re: ***Spoiler*** inequality   « Reply #4 on: Jun 24th, 2003, 11:24am » Quote Modify

Okay, I see where you're coming from now. Too non-obvious for me though...
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towr
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 Re: ***Spoiler*** inequality   « Reply #5 on: Jun 24th, 2003, 11:29am » Quote Modify

for those who'd rather not read latex, even though/if they can/could
 « Last Edit: Jun 24th, 2003, 11:30am by towr » IP Logged

Wikipedia, Google, Mathworld, Integer sequence DB
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