# Plotting partial sums and its Cesàro means

I have the function as shown in the first line of the code and its partial sum in the second line of the code below.

f[x_] := Abs[x];
s[k_, x_] := (Pi/2) + Sum[((-1)^(n) - 1) 2/(Pi n^(2)) Cos[n x], {n, 1, k}]


Then I have the Cesàro means of the function:

$$F_{n}(f)=\frac{1}{n} \sum^{n-1}_{m=0} S_{m}(f)$$

Originally I plot just the function and its partial sum with the code

Plot[Evaluate[{f[x], partialsums}], {x, -0.5, 0.5}


Now if I want to add in the 4th term of the Cesàro means, may I know how can I do so? I made a few changes and there is always a + symbol appearing inline which I believe is due to some mistake in my program

The original plot without the Cesàro means is

f[x_] := Abs[x];
s[k_, x_] := Pi/2 + Sum[((-1)^(n) - 1) 2/(Pi n^(2)) Cos[n x], {n, 1, k}]
partialsums = Table[s[n, x], {n, 4}];
Plot[Evaluate[{f[x], partialsums}], {x, -0.5, 0.5}]


When I introduce the Piecewise function to add the Cesàro means to my graph, no drawings were generated. Please help.

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First, we need to straighten out some of the notation. Whereas your partial sum function s depends only upon the index k and the variable x but not the function f, your list partialsums does not depend on x and your Cesaro partial-sum function, as you expressed it in $\text{LaTeX}$ form, includes an explicit dependence on f. Let's omit the f on the last and do include the x with partialsums. Then:

f[x_] := Abs[x]
s[k_, x_] :=
Pi/2 + Sum[((-1)^(n) - 1) 2/(Pi n^(2)) 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, -0.5, 0.5}]


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you didn't define the cesaro means in the code. So Mathematica can't generate a graph for it. –  Sandra Jan 9 '13 at 16:46
@Sandra: I fixed that omission. BTW, did you know you can directly embed LaTeX code in questions here, by enclosing within dollar signs? –  murray Jan 9 '13 at 16:49
Thanks Murray. Oh I didn't know about that. In my post, I think VLC edit it for me. Now when I graph your code, I do receive the function line and the 4th term of the partial sum line but for the 4th term of the Cesaro means there are two lines generated. Can you have a check on what is wrong in the code? –  Sandra Jan 9 '13 at 16:52
When I put in the legends, it seem not match up: PlotLegends -> {"f(x)=x", "Fourier, 4 terms", "Cesaro, 4 terms"}, PlotStyle -> {Blue, Dashed, Red}] –  Sandra Jan 9 '13 at 17:07
@sandra Since Murray is using partialsums in the plot which is a list with 4 functions you get 6 plots in total. You don't see all of them because there are two twin function involved. The printed functions you get in the output are caused by a missing ; in the code, or a definition using = instead of a :=. –  Sjoerd C. de Vries Jan 9 '13 at 19:33