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I have a following series

$$ \sum\frac{1}{n^2+m^2} $$

As far as I understand it converges. I tried Cauchy criteria and it showed divergency, but i may be mistaken.

When I calculate it in matlab or Maxima it have a good behaviour and converge to finite number about 10.17615092535112.

The convergency plot is following:

enter image description here

  • 0
    Over which $n$ and $m$ do you take the sum? All $n$, $m$? Are they independent? I think then the series diverges...2012-05-24
  • 1
    Just to make sure (since I have no experience with matlab etc.) what does it give you for the series $\sum\frac 1n$?2012-05-24
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    thanks you, Simon, $\frac 1n$ also do not go to infinity and stay at really low values, like 12.09 at $10^5$ terms. That very strange.2012-05-24
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    That's what I expected. This has something to do with machine accuracy. Your computer can only represent numbers down to a certain number of digits. Hence for your computer the harmonic series is finite.2012-05-24
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    I bet you can even "show" that the harmonic series is constant after finitely many steps2012-05-24
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    I checked in matlab: $1/n$ diverges and 2D case $1/(n^2 + m^2)$ converges.2012-05-24
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    yep, this is something with machine accuracy, but nevertheless, those series have "logarithmic" divergency speed.2012-05-24
  • 0
    See also: [How to prove $\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{1}{m^2+n^2}=+\infty$](https://math.stackexchange.com/q/851302)2017-12-10

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