3
$\begingroup$

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

4

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
    @MathGems: Thanks! You can still do, I suppose :-)2013-04-13