5
$\begingroup$

Possible Duplicate:
Is there an elementary proof that $∑_{k=1}^n 1/k$ is never an integer?

Hello,

Prove that $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n}$ is not an integer.

I tried to prove by induction on $n$, but I was stuck :(

Assume $1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n} = \frac{a}{b}$ for some integers $a, b$ and $a \neq b \text{and} b \neq 0$
Then $ 1 + \frac{1}{2} + \frac{1}{3} + ... + \frac{1}{n + 1} = \frac{a}{b} + \frac{1}{n + 1}$

Then how can I prove that this expression is not integer? A hint would be greatly appreciated.

Thanks,
Chan

  • 0
    @Jacob: No. Although not explicitly stated in the question, the subsequent remarks on trying to prove it make it clear that Chan wants to show that it is never an integer when $n\geq 2$, not just there there exists an $n$ such that it is not an integer.2011-02-28

2 Answers 2

5

Hint: look at the largest power of 2 less than $n$. Can it get canceled out from the denominator?

9

HINT: There is always a prime between $\frac{n}{2}$ and $n$, $\forall n \geq 4$

  • 13
    That's overkill.2011-02-28