The harmonic number $H_n$ is defined as $$H_n = \frac11 + \frac12 + \cdots + \frac1n$$ I need to show that $H_n - H_k$ can't be a Natural number, when $n>k$. I understand that the sum is greater than $\ln\big(\frac{n}{k}\big)$ but I don't know how it helps.
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Welcome to MSE. You're right in that the estimate doesn't help at all. This is a number theory question, no calculus involved. – saulspatz Apr 19 '20 at 18:00
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Same question here. – Dietrich Burde Apr 19 '20 at 18:12
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Here's a couple of hints. We want to evaluate $$\sum_{j=k+1}^n\frac1j$$ Let $s$ be the greatest integer such that $2^s$ divides one of the numbers $k+1,k+2,\dots,n$.
First, show that $2^s$ divides exactly one of these numbers.
Then, suppose $$\sum_{j=k+1}^n\frac1j=N\tag1$$ for some integer $N$.
Multiply both sides of $(1)$ by $2^{s-1}$ and observe that exactly one of the terms has an even denominator. Deduce a contradiction. (Hint: what can you say about the sum of two fractions with odd denominators?
saulspatz
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