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 Post subject: Fun with summations (1)Posted: Thu, 25 Oct 2007 17:53:54 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
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Location: Austin, TX
Evaluate the following:

for 0<x<1 and is the greatest integer function.

where s(n) is the number of 1s in the binary expansion of n.

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 Post subject: Posted: Sun, 4 Jan 2009 16:12:30 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7624
Location: NCTS/TPE, Taiwan
I recognise the first one at sight...
Spoiler:
The (-1)^(...) suggests the Haar wavelets... and once you get that it isn't long to get 1-2x as the function with the appropriate "Fourier" coefficients. Finally, right-continuity together with the values at diadic rationals force 1-2x as the required sum.

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 Post subject: Posted: Tue, 6 Jan 2009 15:11:37 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7624
Location: NCTS/TPE, Taiwan
And for the second one:
Spoiler:
Convert to double sum by splitting up s(n) into places where there are 1, then swap the order of sum (since all summands are nonnegative), so the contribution from the 2^k-place 1s evaluates to (1/2^k)log(2), from 1/(n(n+1))=1/n-1/(n+1) and the well known alternating series 1-(1/2)+(1/3)-(1/4)+...=log(2). So the result is 2*log(2).

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