Plotting discrete data but not using discreteplot function

For a given function:

Plot[Sqrt[Abs[x]], {x, -Pi, Pi}]

I have the code to draw the function (with its Abs remove), partial sums and cesaro means as:

f[x_] := Sqrt[x]
s[k_, x_] := \frac{2\sqrt{\pi}}{3}+(-Sqrt[2] FresnelS[Sqrt[2] Sqrt[n]] + 2 Sqrt[n] Sin[n \[Pi]])/(n^(3/2) Sqrt[\[Pi]]) Cos[n x], {n, 1, k}] 
partialsums[x_] = Table[s[n, x], {n, {4}}]; 
c[n_, x_] := (1/n) Sum[s[m, x], {m, 0, n - 1}]

Plot[Evaluate[{f[x], partialsums[x], c[4, x]}], {x, -Pi, Pi},
PlotLegends -> {"f(x)=x", "Fourier, 4 terms", "Cesaro, 4 terms"}, 
PlotStyle -> {{Blue}, {Dashed, Thickness[0.006]}, {Red, Thickness[0.006]}}]

This code fails on my computer and hence I resolve to manual computation.

Updates: It turn out that I can easily solve this issue by removing the k with any number rather than letting it to be indefinite. Although I am not certain the graph is right for k=4 as both graphs(Partial and Cesaro) coincides with each other.

Answer

Maybe

plt = Plot[f[k], {k, 0, 50}, Frame -> True, PlotStyle -> Red,ImageSize -> 300];
dplt = DiscretePlot[cesaro[k], {k, 0, 50}, Frame -> True, PlotRange -> PlotRange[plt],

PlotStyle -> Directive[{Blue, Dashed}], Joined -> True, ImageSize -> 300]; 
Row[{plt, dplt, Show[plt, dplt]}]

enter image description here

Update: or, perhaps, this:?

 dplt2 = DiscretePlot[cesaro[k], {k, 0, 50}, Frame -> True, Filling -> None, 
 PlotRange -> PlotRange[plt], PlotStyle -> Blue, Joined -> True, ImageSize -> 300];
 Row[{plt, dplt2, Show[plt, dplt2]}]

enter image description here

or, using Interpolation on cesaro[k] and

 intFCsr = Interpolation[Table[{k, cesaro[k]}, {k, 0, 50}]];
 Plot[{f[k], intFCsr[k]}, {k, 0, 50}, Frame -> True,PlotStyle -> {Red, Blue}]

enter image description here

Update 2:

 intFCsr = Interpolation[Table[{k, cesaro[k]}, {k, 0, 50}]];
 intFPrtlSms = Interpolation[Table[{k, part[k]}, {k, 0, 50}]];
 Plot[{f[k], intFCsr[k], intFPrtlSms[k]/15}, {k, 0, 50},  ImagePadding -> 45,
 Frame -> True, PlotStyle -> {Red, Blue, Black}, ImageSize -> 500,

 FrameLabel -> {{Style["f, cesaro", 12], Style["partial sum", 12]},
   {Style["k", 12], Style["plot label", 14]}},
 FrameTicks -> {{Join[{#, #, {.01, 0}} & /@ Range[0, 4.],
   {#, " ", {.0075, 0}} & /@ Range[0.2, 4., .2]],
  Join[{#, 15 #, {.01, 0}} & /@ Range[0, 4.],
    {#, " ", {.0075, 0}} & /@ Range[0.2, 4., .2]]}, {Automatic, Automatic}}]

enter image description here

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