This notebook plots the Fourier Sine Series of f(x) = (Sqrt[x]) on the interval (0, 1).  Note that f(x) is NOT piecewise smooth on (0, 1), so we don't have a theorem that says that the Fourier Sine Series converges.  On the other hand, the theorem does not say that the FSS diverges.

L = 1;
f[x_] = (Sqrt[x]);
(*Sine Series coefficients in a list*)
B = Table[(2/L)NIntegrate[f[x]Sin[n π x/L], {x, 0, L}], {n, 100}];
Manipulate[Plot[{f[x], (∑_(n=1)^nTerms)B[[n]] Sin[n π x/L]}, {x, -0.5L, 1.5L}, PlotRange -> {-1.18,1.18}], {{nTerms, 20}, Range[1,100], Appearance -> "Labeled"}, ControlType -> Slider, SaveDefinitions -> True]

It appears that the FSS converges pointwise.