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  • Empty
Does this sum have a closed form?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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  • Empty
\[\sum_{n=0}^\infty (2n+1) e^{-k(n^2+n)} \] where k > 0 is a real number.
amistre64
  • amistre64
when i take the integral, i get something close to the closed form.
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  • Empty
Hmmm, how could I go about finding the error of the approximation? I am fine with this idea this sounds pretty good.

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amistre64
  • amistre64
\[c_n~e^{-k(n^2+n)}\] gives me the 1/e stuff, but my coefficients are not proper ...
amistre64
  • amistre64
i wouldnt know off hand how to find the error stuff. its been a long day
amistre64
  • amistre64
k=1; 3,10,21,36,55,78,... k=2; 3,10,21,36,55,78,... k=1/2; same progresion
amistre64
  • amistre64
3,10,21,36,55,78 7 11 15 19 23 4 4 4 4 3+7n+4n(n-1)/2 seems to cover that ... but thats starting at n=0 so a shift is needed
amistre64
  • amistre64
heh, 2n^2+n is the simplified shift ..
Empty
  • Empty
Yeah that's totally fine, I was thinking more or less of solving them independently to within a constant or something I'm not too concerned this is for my own interest in looking for the closed form of the molecular partition function of a linear rigid rotor. I noticed it doesn't have a closed form and the book I'm reading says to evaluate it numerically but I thought that might be kind of lame and wanted a closed form haha.
amistre64
  • amistre64
with a little help from the wolf tech ... \[\sum_{n=0}^{\infty} (2n+1)e^{-k(n^2+n)} =(2n^2+n) e^{-k(n^2+n)}\] it works for a couple of test values for k
amistre64
  • amistre64
http://www.wolframalpha.com/input/?i=table[%282n^2%2Bn%29e^%28-2%28n^2%2Bn%29%29%2C{n%2C10}] http://www.wolframalpha.com/input/?i=table[sum%28i%3D1+to+n%29+%282n%2B1%29e^%28-%28n^2%2Bn%29%2F2%29%2C{n%2C10}]
amistre64
  • amistre64
firefox doesnt like to paste the links correctly ..
amistre64
  • amistre64
i notice the shift in the polynomial part, like a usual difference equation. \[(1+2n) +0n^2+0n^3+...\color{red}{\implies} 0+(1n+2n^2)+0n^3+...\]
amistre64
  • amistre64
not sure how useful my prattling is, but it looked fun to try out is all :)
amistre64
  • amistre64
the setup doesnt work for n=0 tho ...
Empty
  • Empty
That's fine I think this helped actually haha thanks xD
thomas5267
  • thomas5267
Mathematica returned nothing useful. Looks like it doesn't have a close form.
amistre64
  • amistre64
i noticed an error in my plans ... my table indexed i, from 0 to n .... but i forgot to use i in the process :/ oh well. it still looks fun tho

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