n\}$ ($n\in\Bbb N$); find $X\cap\Bbb N$

Asked Oct 01 '16 at 02:54

Active Oct 01 '16 at 05:53

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Let $X=\{x\mid x=1+1/2+1/3+1/4+\cdots+1/n\}$ where $n\in\Bbb N$. Find $X\cap \Bbb N$.

What I can figure out is to use a general formula and then apply divisibility theorems to solve this, but I suppose there is no way to find the sum of a harmonic series. Can anybody give me a hint as to what other methods can be used to solve this question?

edited Oct 01 '16 at 05:53

Michael Hardy

asked Oct 01 '16 at 02:54

oshhh

2,632

2

HINT: Show that the only harmonic number that is an integer is $H_1=1$. To do this, show that if $n>1$, and $2^k$ is the highest power of $2$ dividing any positive integer $\le n$, then $2^{k-1}H_n$ cannot be an integer and hence neither can $H_n$. – Brian M. Scott Oct 01 '16 at 03:16
@KajHansen It is! Thanks :) – oshhh Oct 01 '16 at 03:16

$X=\{x\mid x=1+1/2+1/3+1/4+\cdots+1/n\}$ ($n\in\Bbb N$); find $X\cap\Bbb N$

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