Plotting a graph with conditions

Question

I am a newbie in Mathematica and is currently using Mathematica v9.0. I would like to consult if anyone knows how can I plot a given functions with conditions as shown below on how they look like.

$$ g(x)= \left\{\begin{split} \frac{(\pi-x)}{2} &\text{, } 0 \leq x \leq \pi \\ -\frac{(\pi+2)}{2} &\text{, } -\pi \leq x < 0 \end{split}\right. $$

Then together in the same graph, I would like to plot its partial sums $S_{4}$ and $S_{8}$ based on the fourier series of the function $\sum^{n}_{k=1} \frac{\sin kx}{k}$.

Note: If there is anyone who can edit the above latex to mathematics form, please do so as I don't know how to convert the above $\LaTeX$ code into mathematical wordings. Sorry about that.

Answer 1

Since there still seems to be no answer to the second part about adding the partial sums to the plot, here you go:

partialSum[n_, x_] := Sum[Sin[k x]/k, {k, n}]
Plot[{
  Piecewise[{{(Pi - x)/2, 
     0 <= x <= Pi}, {-((Pi + x)/2), -Pi <= x <= 0}}],
  partialSum[4, x], partialSum[8, x]},
 {x, -2, 2}]

As you see, you can show several functions in the same plot just by passing a list of functions to Plot. The definition of partialSum is pretty straight forward, too.

Btw, if you don't know the Fourier series already, you can calculate it easily (using FourierSinSeries in this case to immediately get the desired form):

g[x_] := 
 Piecewise[{{(Pi - x)/2, 
    0 <= x <= Pi}, {-((Pi + x)/2), -Pi <= x <= 0}}]

FourierSinSeries[g[x], x, 4]

(* -----> Sin[x] + 1/2 Sin[2 x] + 1/3 Sin[3 x] + 1/4 Sin[4 x] *)

Answer 2

I had solved the first part to my question. Thanks to @b.gatessucks

Plot[Piecewise[{{(Pi - x)/2, 0 <= x <= Pi}, {-((Pi + x)/2), -Pi <= x <= 0}}], {x, -2, 2}]