# Let S_{N}(x)=frac{4}{pi} sum_{n=1}^{N} frac{1 - (-1)^{n}}{n^{3}} sin(nx). Construct graphs of S_{N}(x) and x(pi - x), for 0 leq x leq pi, for N=2 and

Let
Construct graphs of
This will give some sense of the correctness of Fourier’s claim that this polynomial could be exactly represented by the infinite series
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Layton

Step 1
Note that for even n the coefficient in the sum is zero, so some terms are absent.

Step 2

Step 3
We see that ${S}_{10}\left(x\right)$ is a very good a approximation of the function

The absolute value of their difference is given below for the comparison: