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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.... · 3 years, 11 months ago

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

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

$\infty \not\in \mathbb{N} \implies \infty \not\in \mathbb{Z}.$ · 3 years, 11 months ago