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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 $$r>0$$

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Isn't $\int^{n+1}_1 x^{-r}dx$ sufficient for your purposes? – Pietro Majer Apr 27 '12 at 11:24
The tails of such sums are easily bounded by integrals. – Charles Matthews Apr 27 '12 at 11:25
A good question, but not of research interest (unless you want bounds better than the best currently known). This website is for research; try – Gerry Myerson Apr 27 '12 at 12:45
@charles: what do you mean by tails? These are all divergent... – Igor Rivin Apr 27 '12 at 15:38
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)? – user9072 Apr 27 '12 at 17:14

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