Sum of reciprocal of the terms of an arithmetic progression:

Let \(a\) and \(d\) be two coprime integers.
(The great common divisor between \(a\) and \(d\) is \(1\)).
Prove that: \((\forall n\in\mathbb{N}^*):\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+\cdots+\frac{1}{a+nd}\notin\mathbb{Z}\).

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## Comments

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TopNewestYou could edit title as Sum of an Harmonic Progression....

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Why is \(\gcd(a,d)=1\)? If this holds, then it also holds for \(a,d\) with \(\gcd(a,d)>1\).

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What about \(a=1\), \(d=1\), and \(n=\infty\)? Or is that not allowed? EDIT: would \(\infty\in\mathbb{Z}\)?

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\[ \infty \not\in \mathbb{N} \implies \infty \not\in \mathbb{Z}. \]

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