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Prove (or disprove) the following statement: For any positive integers $x,y,t$,

$\displaystyle\sum_{i=1}^{t(y+1)-1} \frac{1}{t(xy+x-1)-x+i}$

is an increasing function of $t$.

My attempts: The statement appears to be true numerically. Tried some obvious bounds to compare the sums for consecutive values of $t$ but didn't find one that was strong enough to prove the statement.

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1 Answer 1

You should be able to use the fact that the $n^{th}$ Harmonic Number

$H_n = \ln n + \gamma + \frac{1}{2n} - O(\frac{1}{n^2})$

Your sum is a difference of two such numbers and so is approximately of the form $\ln\frac{at+b}{ct+d}$ where $a > c$.

Sorry, haven't done the complete math, but this approach looks promising.

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Well hey there M :) –  BlueRaja - Danny Pflughoeft Aug 12 '10 at 17:47
@BlueRaja: Hey! :-) Just discovered this one today. Finally something to get rid of my addiction to stackoverflow :-P –  Aryabhata Aug 12 '10 at 18:06
Hi from me too. :-) Another future addiction for you, given your answers on SO: you may want to commit on the proposal for theoretical computer science (rm "referrer" part of the link if you wish). –  ShreevatsaR Aug 13 '10 at 6:13
@Shree: Hi, I never knew that existed! Thanks for pointing me to that. –  Aryabhata Aug 13 '10 at 7:37
@Aryabhata Welcome back! I miss pinging your old name... –  Math Gems Apr 12 '13 at 23:16

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