Difference between two harmonic numbers when n is not infinite

Question

I need to compute the difference between two harmonic number, in particular :

$H_n - H_{n-pn}$

and i see in this answer Calculating/Estimating difference between Harmonic numbers that $H_n = ln(n) + C + o(1)$ where C is the Euler-Mascheroni constant and the assumption is that n goes to infinity. If i doesn't have the last condition (n infinity) and $0<p<1$, can i compute that difference in this way? or how can i do? Thanks

score 1 · Answer 1 · answered Jan 10 '14 at 17:09

1

An exact expression? No chance, already the $H_n$s do not have one. Asymptotics? The expansion you mentioned yields $\lim\limits_{n\to\infty}H_n-H_{n-pn}=-\log(1-p)$.

answered Jan 10 '14 at 17:09

Did

279,727

ok, but can i say that $H_n - H_{n-pn} \leq ln(n) - ln(n-pn) + O(1)$ where O(1) is a costant <= 1 ?? – user3158123 Jan 10 '14 at 17:17
it is just an example because i have to proof that this difference has value no more than $1+ ln(\frac{1}{1-p})$ – user3158123 Jan 10 '14 at 17:21
If this is what you want to know, I am afraid you will need to ask a new question... – Did Jan 10 '14 at 17:45

Difference between two harmonic numbers when n is not infinite

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