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edit approved Feb 13, 2017 at 11:43

Your sum doen't converge at all. Look how your summands behave if you go to infinity with series. Here the m-sum, postivie and negative added:

    serm1 = 
    Series[m Log[
    2 r - r Cos[\[Theta]]Cos[θ] + m (Cos[\[Theta]]Cos[θ] - 1) + 
    n Sin[\[Theta]]Sin[θ] + m^2 + n^2] - 
    m Log[2 r - r Cos[\[Theta]]Cos[θ] - m (Cos[\[Theta]]Cos[θ] - 1) + 
    n Sin[\[Theta]]Sin[θ] + m^2 + n^2], {m, Infinity, 1}] // Normal

     (*   -2 + 2 Cos[\[Theta]]  *)

Every step you add a constant value, therefore your sum must go to infinity. It's even worse with the n-Sum:

    sern1 = 
     Series[m Log[
     2 r - r Cos[\[Theta]]Cos[θ] + m (Cos[\[Theta]]Cos[θ] - 1) + 
     n Sin[\[Theta]]Sin[θ] + m^2 + n^2] + 
     m Log[2 r - r Cos[\[Theta]]Cos[θ] + m (Cos[\[Theta]]Cos[θ] - 1) - 
     n Sin[\[Theta]]Sin[θ] + m^2 + n^2], {n, Infinity, 1}] // Normal

     (*   -4 m Log[1/n]  *)
 
     Limit[sern1, n -> Infinity]

     (*  m \[Infinity]  *)

You can calculate a partial sum numericaly (analytical integration didn't work) to have an impression it goes to infinity:

     nint[r_, m_, n_] := 
     NIntegrate[
     Log[2 r - r Cos[\[Theta]]Cos[θ] + m (Cos[\[Theta]]Cos[θ] - 1) + 
     n Sin[\[Theta]]Sin[θ] + m^2 + n^2], {\[Theta]θ, 0, 2 Pi}] 


  nsum[r_, m0_, n0_] := 
  NSum[m nint[r, m, n] + m nint[r, m, -n], {n, 1, n0}, {m, -m0, m0}]

  Table[nsum[1, k, k], {k, 1, 10}]

   )*  {-6.10232, -34.2553, -88.7863, -169.822, -277.058, -410.236, \
         -569.171, -753.729, -963.815, -1199.36}   *)

Your sum doen't converge at all. Look how your summands behave if you go to infinity with series. Here the m-sum, postivie and negative added:

    serm1 = 
    Series[m Log[
    2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
    n Sin[\[Theta]] + m^2 + n^2] - 
    m Log[2 r - r Cos[\[Theta]] - m (Cos[\[Theta]] - 1) + 
    n Sin[\[Theta]] + m^2 + n^2], {m, Infinity, 1}] // Normal

     (*   -2 + 2 Cos[\[Theta]]  *)

Every step you add a constant value, therefore your sum must go to infinity. It's even worse with the n-Sum:

    sern1 = 
     Series[m Log[
     2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
     n Sin[\[Theta]] + m^2 + n^2] + 
     m Log[2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) - 
     n Sin[\[Theta]] + m^2 + n^2], {n, Infinity, 1}] // Normal

     (*   -4 m Log[1/n]  *)
 
     Limit[sern1, n -> Infinity]

     (*  m \[Infinity]  *)

You can calculate a partial sum numericaly (analytical integration didn't work) to have an impression it goes to infinity:

     nint[r_, m_, n_] := 
     NIntegrate[
     Log[2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
     n Sin[\[Theta]] + m^2 + n^2], {\[Theta], 0, 2 Pi}]

  nsum[r_, m0_, n0_] := 
  NSum[m nint[r, m, n] + m nint[r, m, -n], {n, 1, n0}, {m, -m0, m0}]

  Table[nsum[1, k, k], {k, 1, 10}]

   )*  {-6.10232, -34.2553, -88.7863, -169.822, -277.058, -410.236, \
         -569.171, -753.729, -963.815, -1199.36}   *)

Your sum doen't converge at all. Look how your summands behave if you go to infinity with series. Here the m-sum, postivie and negative added:

    serm1 = 
    Series[m Log[
    2 r - r Cos[θ] + m (Cos[θ] - 1) + 
    n Sin[θ] + m^2 + n^2] - 
    m Log[2 r - r Cos[θ] - m (Cos[θ] - 1) + 
    n Sin[θ] + m^2 + n^2], {m, Infinity, 1}] // Normal

     (*   -2 + 2 Cos[\[Theta]]  *)

Every step you add a constant value, therefore your sum must go to infinity. It's even worse with the n-Sum:

    sern1 = 
     Series[m Log[
     2 r - r Cos[θ] + m (Cos[θ] - 1) + 
     n Sin[θ] + m^2 + n^2] + 
     m Log[2 r - r Cos[θ] + m (Cos[θ] - 1) - 
     n Sin[θ] + m^2 + n^2], {n, Infinity, 1}] // Normal

     (*   -4 m Log[1/n]  *)
 
     Limit[sern1, n -> Infinity]

     (*  m \[Infinity]  *)

You can calculate a partial sum numericaly (analytical integration didn't work) to have an impression it goes to infinity:

     nint[r_, m_, n_] := 
     NIntegrate[
     Log[2 r - r Cos[θ] + m (Cos[θ] - 1) + 
     n Sin[θ] + m^2 + n^2], {θ, 0, 2 Pi}] 


  nsum[r_, m0_, n0_] := 
  NSum[m nint[r, m, n] + m nint[r, m, -n], {n, 1, n0}, {m, -m0, m0}]

  Table[nsum[1, k, k], {k, 1, 10}]

   )*  {-6.10232, -34.2553, -88.7863, -169.822, -277.058, -410.236, \
         -569.171, -753.729, -963.815, -1199.36}   *)

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created Feb 13, 2017 at 11:20

Akku14

17.4k
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Your sum doen't converge at all. Look how your summands behave if you go to infinity with series. Here the m-sum, postivie and negative added:

    serm1 = 
    Series[m Log[
    2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
    n Sin[\[Theta]] + m^2 + n^2] - 
    m Log[2 r - r Cos[\[Theta]] - m (Cos[\[Theta]] - 1) + 
    n Sin[\[Theta]] + m^2 + n^2], {m, Infinity, 1}] // Normal

     (*   -2 + 2 Cos[\[Theta]]  *)

Every step you add a constant value, therefore your sum must go to infinity. It's even worse with the n-Sum:

    sern1 = 
     Series[m Log[
     2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
     n Sin[\[Theta]] + m^2 + n^2] + 
     m Log[2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) - 
     n Sin[\[Theta]] + m^2 + n^2], {n, Infinity, 1}] // Normal

     (*   -4 m Log[1/n]  *)
 
     Limit[sern1, n -> Infinity]

     (*  m \[Infinity]  *)

You can calculate a partial sum numericaly (analytical integration didn't work) to have an impression it goes to infinity:

     nint[r_, m_, n_] := 
     NIntegrate[
     Log[2 r - r Cos[\[Theta]] + m (Cos[\[Theta]] - 1) + 
     n Sin[\[Theta]] + m^2 + n^2], {\[Theta], 0, 2 Pi}]

  nsum[r_, m0_, n0_] := 
  NSum[m nint[r, m, n] + m nint[r, m, -n], {n, 1, n0}, {m, -m0, m0}]

  Table[nsum[1, k, k], {k, 1, 10}]

   )*  {-6.10232, -34.2553, -88.7863, -169.822, -277.058, -410.236, \
         -569.171, -753.729, -963.815, -1199.36}   *)