Plotting a series

Question

I'm trying to find a way to plot the sum of a series from n to nmax.

Here is the code for the series:

ClearAll[a, b, Gx, Gy, n, bD, nmax, simp]
simp[n_] = 
  Simplify[1/(3 Sqrt[Gx] (π + 2 n π)^5)
     8 Gy (b Sqrt[
       Gx] (1 + 2 n) π (a^3 (π + 2 n π)^4 + 
         96 (-1)^n Sin[1/2 a (1 + 2 n) π]) + 
      96 (-1)^n Sqrt[
       Gy] (a (1 + 2 n) π Cos[1/2 a (1 + 2 n) π] - 
         4 Sin[1/2 a (1 + 2 n) π]) Tanh[(
        b Sqrt[Gx] (1 + 2 n) π)/(2 Sqrt[Gy])])];
bD[nmax_] := Sum[simp[n], {n, 0, nmax}];
a = 0.005; b = a; Gy = 41018756.0; Gx = 72463203.0; nmax = 200;

I tried Plot[Evaluate[Table[bD[nmax], {n, 5}]], {n, 0, 5}], but it produces a straight line. It seems like most times when others had similar problems, like here and here, they had two constants to work with. I only have n so I'm not sure how to proceed.

When I check the contents of [Table[bD[nmax], {n, 5}]], I find that it has {0.26291, 0.26291, 0.26291, 0.26291, 0.26291}, so I'm guessing that for some reason that I can't see, Mathematica is doing the full sum for every entry in the table.

Edit

I managed to get a plot using ListPlot[Table[Sum[simp[n], {n}], {n, 0, nmax}]], but comparing the last value in the table to that from Sum[simp[n], {n, 0, nmax}] shows that they are different. Using the table seems to give a different result to just using a sum.

m_goldberg · Accepted Answer · 2019-04-01T08:56:15.810

Let's redefine the terms of your sum so that they evaluated fairly quickly.

With[{a = 0.005, b = 0.005, Gy = 41018756.0, Gx = 72463203.0}, 
  term[n_] = 
    1/(3 Sqrt[Gx] (π + 2 n π)^5) 8 Gy * 
      (b Sqrt[Gx] (1 + 2 n) π (a^3 (π + 2 n π)^4 + 
        96 (-1)^n Sin[1/2 a (1 + 2 n) π]) + 96 (-1)^n Sqrt[Gy] * 
          (a (1 + 2 n) π Cos[1/2 a (1 + 2 n) π] - 4 Sin[1/2 a (1 + 2 n) π]) * 
            Tanh[(b Sqrt[Gx] (1 + 2 n) π)/(2 Sqrt[Gy])]) // N];

Now we compute a table of 15 terms.

Table[term[i], {i, 15}]

{0.0461359, 0.0816871, 0.0588642, 0.0757376, 0.0623486, 
 0.0734383, 0.0639842, 0.0722128, 0.0649383, 0.0714477, 
 0.0655664, 0.0709223, 0.066013, 0.0705375, 0.0663482}

As you can see, the terms don't vary much, so the accumulated sum will be pretty near a straight line.

sum[nmax_] := Accumulate[Table[term[i], {i, nmax}]]
ListPlot[sum[16]]

I would guess that term[i] is converging to a constant as i -> ∞, and the plot is asymptotically approaching a straight line.

Plotting a series

Edit

1 Answers1