Contrasting amounts of reciprocals

Confirm (or refute) the adhering to declaration: For any kind of favorable integers $x,y,t$,

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

is a raising function of $t$.

My efforts: The declaration seems real numerically. Attempted some noticeable bounds to contrast the amounts for successive values of $t$ yet really did not locate one that was solid sufficient to confirm the declaration.

2019-05-06 21:40:33
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Answers: 1

You need to have the ability to make use of the reality that the $n^{th}$ Harmonic Number

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

Your amount is a distinction of 2 such numbers therefore is about of the kind $\ln\frac{at+b}{ct+d}$ where $a > c$.

Sorry, have not done the full mathematics, yet this strategy looks encouraging.

2019-05-10 07:56:22