Plotting partial sums and its Cesàro means

Question

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.

Answer 1

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}]