Sequence

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

Sequence

Unread post by Anik Islam » Fri Oct 09, 2015 10:32 pm

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
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Re: Sequence

Unread post by Masum » Sat Oct 10, 2015 12:26 am

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

Unread post by Anik Islam » Sat Oct 17, 2015 3:39 pm

Thanks,Masum Bhai

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nayel
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Re: Sequence

Unread post by nayel » Sun Nov 22, 2015 7:12 am

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

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