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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.

1 Answers 1

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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.

  • 0
    Well hey there M :)2010-08-12
  • 2
    @BlueRaja: Hey! :-) Just discovered this one today. Finally something to get rid of my addiction to stackoverflow :-P2010-08-12
  • 0
    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](http://area51.stackexchange.com/proposals/8766/theoretical-computer-science?referrer=5tsJnxHB0YaB49QQOgt7xA2) (rm "referrer" part of the link if you wish).2010-08-13
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    @Shree: Hi, I never knew that existed! Thanks for pointing me to that.2010-08-13
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    @Aryabhata Welcome back! I miss pinging your old name...2013-04-12
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    @MathGems: Thanks! You can still do, I suppose :-)2013-04-13