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Euler's Constant: The asymptotic behavior of $\left(\sum\limits_{j=1}^{N} \frac{1}{j}\right) - \log(N)$

I'm stumped by this one exercise. The question is to "Prove that there exists a positive constant $\gamma$ such that $$\sum_{k=1}^n k^{-1} - \log{n} = \gamma + O(n^{-1})$$ by comparing the sum to a Riemann sum for $\int_1^n x^{-1} dx$."

However, I can't seem to figure out how to show that the approximation is $O(n^{-1})$. By taking the obvious (the left- and right-hand) Riemann sums for the integral, I can get the bound $0 \le \sum_{k=1}^n k^{-1} - \log{n} \le 1$, which is good enough to show that $\gamma = \lim_{n\to \infty} \sum_{k=1}^n k^{-1} - \log{n}$ exists (I think, if I also use the monotonicity of the harmonic numbers and $\log$), but not that the error is $O(n^{-1})$. I think that I need to use the definition of $\gamma$ to get the approximation, but I don't see how I can do any more with the Riemann sum.

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marked as duplicate by Eric Naslund, Sivaram Ambikasaran, Mike Spivey, GEdgar, Sasha Dec 19 '11 at 22:49

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

This is certainly a duplicate, I just can't find it. – Eric Naslund Dec 19 '11 at 21:43
That's what I thought (it seems like the first question to ask about the constant), but I couldn't find it either. – Calvin McPhail-Snyder Dec 19 '11 at 21:49

1 Answer 1

up vote 5 down vote accepted

Let $[x]$ denote the floor of $x$, and $\{x\}$ the fractional part. Then $$\sum_{k=1}^{n} \frac{1}{k} =1+\int_1^n \frac{[x]}{x^2}dx.$$ Then since $$\int_1^n \frac{[x]}{x^2}dx=\log n -\int_1^n \frac{\{x\}}{x^2}dx, $$ we have $$ \sum_{k=1}^{n} \frac{1}{k}= \log n +1-\int_1^n \frac{\{x\}}{x^2}dx.$$

Now, $$\int_1^n \frac{\{x\}}{x^2}dx=\int_1^\infty \frac{\{x\}}{x^2}dx+O\left(\frac{1}{n}\right),$$ so we get the desired result.

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