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 Post subject: strange limit repost
PostPosted: Wed, 10 Dec 2008 01:16:00 UTC 
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Joined: Sat, 16 Aug 2008 04:47:19 UTC
Posts: 208
Somebody posted a problem on this forum that I haven't been able to find again and was wondering what the answer was....

it was find the limit of the sequence,

\frac{1}{2}, \frac{\frac{1}{2}}{\frac{3}{4}}, \frac{\frac{\frac{1}{2}}{\frac{3}{4}}}{\frac{\frac{5}{6}}{\frac{7}{8}}}, 
\ldots



I used a computer to help me see what it would be but after 2^{32} being in the bottom most term i get the value is something like .707106781186

which looks to me like the limit is \frac{\sqrt{2}}{2}

just wanted to know if there was some trick the poster knew that I couldn't figure out.


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 Post subject:
PostPosted: Wed, 10 Dec 2008 05:56:55 UTC 
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Joined: Sun, 4 May 2003 16:04:19 UTC
Posts: 2906
Good work on getting the answer.

Here is a hint. Consider the limit to be 1^{e_1}2^{e_2}...n^{e_n}.... then {e_n} is related to the Thue-Morse sequence.

_________________
Has anyone noticed that the below is WRONG? Otherwise this statement would be true:
-1\cong1\pmod{13}
i\cong5 \pmod{13} where
i^2=-1


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