0
$\begingroup$

Hi, I need an upper bound for the generalized harmonic number with negative exponents, i.e:

$$H_{n,r}=\sum_{k=1}^n \frac{1}{k^r}$$

where $r<0$ especially, I need a bound for $$r=-\frac{1}{2}$$ Has anyone an idea whether those bounds exist? Thanks!

PS: I've seen the related question, but the answer only deals with $$r>0$$

Improve this question
$\endgroup$
6
  • $\begingroup$ Isn't $\int^{n+1}_1 x^{-r}dx$ sufficient for your purposes? $\endgroup$
    – Pietro Majer
    Commented Apr 27, 2012 at 11:24
  • $\begingroup$ The tails of such sums are easily bounded by integrals. $\endgroup$
    – Charles Matthews
    Commented Apr 27, 2012 at 11:25
  • $\begingroup$ A good question, but not of research interest (unless you want bounds better than the best currently known). This website is for research; try math.stackexchange.com $\endgroup$
    – Gerry Myerson
    Commented Apr 27, 2012 at 12:45
  • 1
    $\begingroup$ @charles: what do you mean by tails? These are all divergent... $\endgroup$
    – Igor Rivin
    Commented Apr 27, 2012 at 15:38
  • 1
    $\begingroup$ It could really help if you provide some context. What type/quality of bound would you need? Or, is your last comment to be understood that the question is no longer relevant (say, since the application you had in mind does not work no matter the bound for this sum)? $\endgroup$
    – user9072
    Commented Apr 27, 2012 at 17:14

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .