I'm using Mathematica to solve an initial value problem, and then plot it. However, When I try and plot it I'm receiving an error message. The problem I'm trying to graph is

Here is my code:

\[Kappa] = 1/4; L = 4; 
  f[x_] = Piecewise[{{100*Sin[Pi*x], 0 <= x < 2}, {0, 2 <= x < 4}}]
Plot[f[x], {x, 0, 4}, AxesLabel -> Automatic, PlotRange -> Full]
un[x_, t_, n_] = Exp[-1/4*(n Pi/4)^2 t] Sin[n Pi x/4]
an[n_] = 2/4 Integrate[f[x]*Sin[n Pi x/4], {x, 0, 4}]
an[n_] = Simplify[an[n], n \[Element] Integers]
u[x_, t_, m_] = Sum[an[n] un[x, t, n], {n, 1, m}]
Plot[{f[x], u[x, 0, 1]}, {x, 0, L}, AxesLabel -> Automatic, 
 PlotLegends -> "Expressions"]
Plot[{f[x], u[x, 0, 20]}, {x, 0, L}, AxesLabel -> Automatic, 
 PlotLegends -> "Expressions"]

The last part of my code, where I am trying to change the value of m is where I running into issues. How can I fix this? Thanks!