## Sequence

For discussing Olympiad Level Number Theory problems
Anik Islam
Posts: 2
Joined: Fri Aug 21, 2015 9:17 pm

### Sequence

As-salamu alaykum...........I have a question....

$1+\dfrac12+\dfrac13+\ldots=?$
Last edited by Masum on Sat Oct 10, 2015 12:24 am, edited 1 time in total.
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Masum
Posts: 592
Joined: Tue Dec 07, 2010 1:12 pm

### Re: Sequence

This can be written as $\zeta(1)$ (don't be afraid if you never saw this, search for Zeta Function).
$1+\dfrac12+\dfrac13+\ldots=\prod_{p\in\mathbb P}\dfrac{p}{p-1}$
where $\mathbb P$ is the set of primes.
Why does this hold? Think using unique prime factorization.
One one thing is neutral in the universe, that is $0$.

Anik Islam
Posts: 2
Joined: Fri Aug 21, 2015 9:17 pm

### Re: Sequence

Thanks,Masum Bhai

nayel
Posts: 268
Joined: Tue Dec 07, 2010 7:38 pm
Location: Dhaka, Bangladesh or Cambridge, UK

### Re: Sequence

It's a divergent series. Neither the infinite sum nor the infinite product has a finite value. So it does not even make sense to say that they are equal. More details: https://en.wikipedia.org/wiki/Harmonic_ ... thematics)
"Everything should be made as simple as possible, but not simpler." - Albert Einstein