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How can I show by induction whether $1+\frac{1}{2}+\frac{1}{3}\cdots+\frac{1}{n}$ is an integer or not?

Progress :
For n=1

the expression is $(=1)$ an integer.
How can I show the next step?

Smart Math
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    The first step, which you have not done, is to clearly express the statement you are trying to prove. Induction proves statements of the type “$P(n)$ is true for all $n$ greater than or equal to $1$”. You don't have a statement of that form, and you can't prove something if you don't know what you are trying to prove/. – MJD Oct 10 '14 at 14:15
  • $H_n$ is never an integer except for $n\in{0,1}$. Usual proof, however, do not use induction. – Jack D'Aurizio Oct 10 '14 at 14:15
  • Welcome to the Math StackExchange. Have you checked n = 2 and n = 3? – Paul Sundheim Oct 10 '14 at 14:15
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    See http://math.stackexchange.com/questions/2746/is-there-an-elementary-proof-that-sum-limits-k-1n-frac1k-is-never-an-int. – lhf Oct 10 '14 at 14:16

1 Answers1

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You can do it by induction, in fact you can prove that $(H_n)$ is always the ratio of an odd number (numerator) and an even number (denominator) :

$H_n = \frac{P_n}{Q_n}$, $P_n$ odd number, $Q_n$ even number

It's doable by induction, you will have to introduce respective powers of two of the numerator and the denominator. It's not very pretty though.

However a more elegant way to prove this fact is to study the biggest power of two that is inferior to n, and to see what it implies for the numerator and the denominator (why $P_n$, when $H_n$ is reduced to an irreductible fraction, has to be an odd number) when you reduce to a common denominator, that is finding the gcd of all (1,2,..,n)

Does it help you?

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