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

Question

Let

S_{N} (x) = \frac{4}{π} \sum_{n = 1}^{N} \frac{1 - {(- 1)}^{n}}{n^{3}} \sin (n x) .

Construct graphs of

S_{N} (x) and x (π - x), f or 0 \leq x \leq π, f or N = 2 and t h e n N = 10 .

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

\frac{4}{π} \sum_{n = 1}^{\infty} \frac{1 - {(- 1)}^{n}}{n^{3}} \sin (n x) o n [0, π] .

Answer 1

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

Step 2
$S_{10} (x) = \frac{8}{π} (\sin x + \frac{1}{27} \sin 3 x + \frac{1}{125} \sin 5 x + \frac{1}{273} \sin 7 x + \frac{1}{729} \sin 9 x)$

Step 3
We see that $S_{10} (x)$ is a very good a approximation of the function
$f (x) = x (π - x) .$
The absolute value of their difference is given below for the comparison:

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

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