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

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

• Look at the power of 2 divisible by numerator and denominator.
– user325
Feb 28, 2011 at 0:24
• Your induction hypothesis is not strong enough, because simply assuming that $k$ is not an integer does not guarantee that $k+\frac{1}{n+1}$ is not an integer. So if you want to proceed by induction, you need to prove more than simply that $H_n=1+\frac{1}{2}+\cdots + \frac{1}{n}$ is not an integer, you need to prove something about its expression as a rational written in lowest terms. Feb 28, 2011 at 0:27
• @Arturo Magidin: Thanks, that was exactly the problem that I encountered, since I actually found counter examples. Feb 28, 2011 at 0:29
• Feb 28, 2011 at 0:31
• Just out of curiosity. Would saying that for $n=2$ the series is not an integer suffice? Proof by example? Feb 28, 2011 at 0:37

HINT: There is always a prime between $\frac{n}{2}$ and $n$, $\forall n \geq 4$
Hint: look at the largest power of 2 less than $n$. Can it get canceled out from the denominator?