Does a Closed Form Exist for the Harmonic Series?

[Dansuer]

HI!

I was wandering if there is a proof that the harmonic sum \sum\frac{1}{k} has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.

 

[Erland]

HI!

I was wandering if there is a proof that the harmonic sum \sum\frac{1}{k} has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.

This series diverges to infinity.

 

[Dansuer]

yeah that's cool but that's not what I'm looking for. Maybe I've not been very clear.
I'll try again.

There is not a closed form expression of the harmonic sum \sum^{n}_{0}\frac{1}{k}. which means it cannot be expressed in terms of elementary functions (e^x,sin(n), log(n), ...).
This is a closed form for \sum^{n}_{0} k

\sum^{n}_{0} k = \frac{n(n+1)}{2}

Does a closed form exist, but it's not yet been found ?
or it had been proved that it cannot exist ?
or maybe there is, maybe not, nobody knows anything about it ?