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Closed form of harmonic series

  1. Oct 31, 2012 #1
    HI!

    I was wandering if there is a proof that the harmonic sum [itex]\sum\frac{1}{k}[/itex] has no closed form. Something like the proof that an equation with degree more than 4 has no solution in terms of radicals.
     
  2. jcsd
  3. Oct 31, 2012 #2

    Erland

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    This series diverges to infinity.
     
  4. Oct 31, 2012 #3
    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 [itex]\sum^{n}_{0}\frac{1}{k}[/itex]. which means it cannot be expressed in terms of elementary functions (e^x,sin(n), log(n), ...).
    This is a closed form for [itex]\sum^{n}_{0} k [/itex]

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

    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 ?
     
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