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Explore the Fourier Series for a square wave

f (t) = (4/pi) Sum[(1/n) sin (2 pi (f (t))), {n, 1, Infinity}]
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closed as off-topic by m_goldberg, corey979, bbgodfrey, Artes, J. M. will be back soon Nov 1 '17 at 2:52

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question arises due to a simple mistake such as a trivial syntax error, incorrect capitalization, spelling mistake, or other typographical error and is unlikely to help any future visitors, or else it is easily found in the documentation." – m_goldberg, corey979, Artes, J. M. will be back soon
If this question can be reworded to fit the rules in the help center, please edit the question.

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Once you have the function and its Fourier representation, you can try with this:

f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, \[Infinity], 2}];
fserf[t_,nmax_] := (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, 2*nmax-1, 2}];
GraphicsGrid[
Partition[
    Plot[{f[t], fserf[t, #]}, {t, 0, 3}, PlotLabel -> ToString[#] <> " Terms"] 
    &/@ {3, 5, 10, 20}, 2],
    ImageSize -> Large]

You can get your plots:

![enter image description here

Even use Manipulateto interactively see how the series works:

f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, \[Infinity], 2}];
fserf[t_, nmax_] := (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, 2*nmax-1, 2}];
Manipulate[
   Plot[{f[t], fserf[t, nmax]}, {t, 0, scale},PlotLabel-> ToString[nmax] <> " Terms", PlotRange -> {-1.5, 1.5}], 
{{nmax, 1, "Nº of Terms"}, 1, 50, 1, Appearance -> "Labeled"},
{{scale, 1, "x-axis length"}, 1, 5, 1, Appearance -> "Labeled"}
]

enter image description here

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First you need to learn the proper syntax for writing Mathematica expressions. When the sum is written properly as

f[t_] = (4/Pi) Sum[(1/n) Sin[2 Pi n t], {n, 1, ∞, 2}]

(2 I (ArcTanh[E^(-2 I π t)] - ArcTanh[E^(2 I π t)]))/π

then the plot is simple:

Plot[f[t], {t, 0, 3}]

plotenter image description here

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