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Prove that the number $$\sum_{k=2}^{n}{1\over k}$$ is not an integer.

symplectomorphic
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Theodoros Mpalis
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1 Answers1

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Suppose that $\sum_{i=2}^n \frac{1}{i}$ is an integer, $n>1$. By Bertrand's postulate, there exists a prime $\frac{n}{2}<p<n$. If we make the common denominator of the fraction, we get $$\sum_{i=2}^n \frac{1}{i} = \frac{1}{n!}\sum_{i=1}^n (2\times 3 \times ... \times (i-1) \times (i+1) \times ... \times n)$$.

Note that $p$ divides the denominator of the above fraction, but in the numerator all terms in the sum except the term $(1\times 2 \times ... \times (p-1) \times (p+1) \times ... \times n)$,and because we have assumed that $\frac{n}{2}<p<n$, this term does not divide $p$ because it can't contain a multiple of $p$. Hence it follows that $p$ does not divide the numerator of the fraction. Hence it follows that the fraction cannot be made an integer for any $n$.

Sarvesh Ravichandran Iyer
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    Does the "$\times$" symbol is the same as"$\cdot$" ? – Theodoros Mpalis Mar 26 '16 at 03:29
  • Yes, both mean multiplication, but the difference is that $\cdot$ often is used for multiplication in groups and rings, while $\times$ is used for multiplying real numbers . By the way, hope you know about Bertrand's postulate. Erdos proved it at the age of nineteen. – Sarvesh Ravichandran Iyer Mar 26 '16 at 03:37
  • Of course. But I thought "$\times$" was used for multiplication in groups and rings and "$\cdot$" for real numbers. Maybe it's oposite in my country... – Theodoros Mpalis Mar 26 '16 at 03:47
  • Your country Greece? Cyprus? Macedonia? All said and done, maybe it is the opposite, but the textbook I use is written by a man from the United Kingdom, while I myself have seen it in use in India. – Sarvesh Ravichandran Iyer Mar 26 '16 at 05:05
  • I'm from Greece my friend – Theodoros Mpalis Mar 26 '16 at 05:56
  • But I think $x\cdot x\cdot\ldots\cdot x\neq x\times x\times\ldots\times x$, lol! – Theodoros Mpalis Mar 26 '16 at 06:02
  • Well, I'm from India, but our notation is British borrowed because we were once their colony. So we use $\times$ whenever there is some relation between $a$ and $b$ i.e. $a \times b = b \times a$ or $a \times b = -b \times a$, while the group product, which has no property other than associativity, is denoted by $\cdot$. I saw it written in I.N.Herstein, a British author. Besides, even $+$ is used, but only for strictly commutative operations. It must be different in Greece though. I have seen variations in papers published in Russia and Slovakia in this regard. – Sarvesh Ravichandran Iyer Mar 26 '16 at 06:03