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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: \((\forall n\in\mathbb{N}^*):\frac{1}{a}+\frac{1}{a+d}+\frac{1}{a+2d}+\cdots+\frac{1}{a+nd}\notin\mathbb{Z}\).

Note by Mountassir Farid
3 years, 11 months ago

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You could edit title as Sum of an Harmonic Progression.... Rahul Nahata · 3 years, 11 months ago

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Why is \(\gcd(a,d)=1\)? If this holds, then it also holds for \(a,d\) with \(\gcd(a,d)>1\). Tim Vermeulen · 3 years, 11 months ago

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What about \(a=1\), \(d=1\), and \(n=\infty\)? Or is that not allowed? EDIT: would \(\infty\in\mathbb{Z}\)? Daniel Liu · 3 years, 11 months ago

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@Daniel Liu \[ \infty \not\in \mathbb{N} \implies \infty \not\in \mathbb{Z}. \] Tim Vermeulen · 3 years, 11 months ago

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