# How to Plot an Infinite Series

I want to sketch the graphs of $$u(t,x)=\frac{1}{2}+\sum_{n\geq 1} \frac{1}{n\pi} ((-1)^n-1)e^{-n^2t}\sin(nx)$$ for $t=0, 0.01, 0.1, 0.5, 1, 10$ on the same axes.

For $t=0$, I input

Plot[{1/2 + Sum[(1/n \[Pi]) ((-1)^n - 1) Sin [n x]], {n, 1, Infinity}}, {x, -\[Pi], \[Pi]}]


Then Mathematica keeps running...

Could you help me with this? Thanks.

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If you define

f[x_, t_, nm_] := 1/2 + Sum[(1/n/ Pi) ((-1)^n - 1) Sin[n x] Exp[-t n^2], {n, 1, nm}];


then

Plot[Table[f[x, t, 150], {t, {0, 0.01, 0.1, 0.5, 1, 10}}] // Release, {x, -Pi, Pi}]


produces

and the "Gibbs ringing" i.e. the small oscillations near the sharp edges come from truncation of the sum at 150 instead of $\infty$.

Treating separately the t=0 case, which can be summed to infinity (see Sjoerd's answer), you can get a pretty accurate plot while choosing nm=1000

 Show[{Plot[
1/2 + Sum[(1/n \[Pi]) ((-1)^n - 1) Sin[n x], {n, 1, Infinity}] //
Evaluate, {x, -\[Pi], \[Pi]},PlotStyle-> Darker[Blue,0.5]],
Plot[Table[
1/2 + Sum[(1/n \[Pi]) Exp[-n^2 t] ((-1)^n - 1) Sin[n x], {n, 1,
1000}], {t, {0.01, 0.1, 0.5, 1, 10}}] //
Evaluate, {x, -\[Pi], \[Pi]}, PlotPoints -> 50]}]


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@J.M. sorry about the Release. I ll try and teach myself. –  chris Nov 4 '12 at 15:27
Thank you so much! –  Vladimir Nov 4 '12 at 15:35
I wasn't objecting to your use of Release[] at all... :) –  Ｊ. Ｍ. Nov 4 '12 at 15:39

Your input contains a syntax error. You put the summation range outside the Sum. Another things that will improve plotting is adding an Evaluate, otherwise Plot will re-calculate the sum for every iteration.

Plot[
1/2 + Sum[(1/n \[Pi]) ((-1)^n - 1) Sin[n x], {n, 1, Infinity}] // Evaluate,
{x, -\[Pi], \[Pi]}
]


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@chris This is the t=0 line of code that he reported keeps on running. For t=0 the Exp can be left away. –  Sjoerd C. de Vries Nov 4 '12 at 15:15