1

Show that if $n, k \in \mathbb{Z}, n>1, k>0$, then $$\frac{1}{n} + \frac{1}{n+1} + \cdots + \frac{1}{n+k}$$ is not an integer.

I thought about showing the result by first showing $k \geq n$ and using the fact that for every $n \in \mathbb{N}$ there is a prime $p$ such that $n <p \leq 2n$, but am not sure if there is another way. If not, how can we proceed and conclude?

EDIT: There is a question like this already asked that I did not see. I do not need assistance with this question.

Dominated Convergence Theorem
  • 923

1 Answers1

2

Let $f(i)$ be the greatest $j\ge 0$ such that $2^j$ divides $i$. Now, there will be always exactly one maximum in the set $$ \left\{f(n),f(n+1),\ldots,f(n+k)\right\}. $$ Hence $\sum_{i\le k}\frac{1}{n+i}$ cannot be an integer.

Paolo Leonetti
  • 15,423
  • 3
  • 24
  • 57