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 then 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}{pi} sum_{n=1}^{infty} frac{1 - (-1)^{n}}{n^{3}} sin(nx) on [0, pi].

Question
Polynomial graphs
asked 2021-01-05
Let \(\displaystyle{S}_{{{N}}}{\left({x}\right)}={\frac{{{4}}}{{\pi}}}\ {\sum_{{{n}={1}}}^{{{N}}}}\ {\frac{{{1}\ -\ {\left(-{1}\right)}^{{{n}}}}}{{{n}^{{{3}}}}}}\ {\sin{{\left({n}{x}\right)}}}.\)
Construct graphs of \(\displaystyle{S}_{{{N}}}{\left({x}\right)}\ {\quad\text{and}\quad}\ {x}{\left(\pi\ -\ {x}\right)},\ {f}{\quad\text{or}\quad}\ {0}\ \leq\ {x}\ \leq\ \pi,\ {f}{\quad\text{or}\quad}\ {N}={2}\ {\quad\text{and}\quad}\ {t}{h}{e}{n}\ {N}={10}.\)
This will give some sense of the correctness of Fourier’s claim that this polynomial could be exactly represented by the infinite series
\(\displaystyle{\frac{{{4}}}{{\pi}}}\ {\sum_{{{n}={1}}}^{{\infty}}}\ {\frac{{{1}\ -\ {\left(-{1}\right)}^{{{n}}}}}{{{n}^{{{3}}}}}}\ {\sin{{\left({n}{x}\right)}}}\ {o}{n}\ {\left[{0},\ \pi\right]}.\)

Answers (1)

2021-01-06
Step 1
Note that for even n the coefficient in the sum is zero, so some terms are absent.
image
Step 2
\(\displaystyle{S}_{{{10}}}{\left({x}\right)}={\frac{{{8}}}{{\pi}}}{\left({\sin{\ }}{x}\ +\ {\frac{{{1}}}{{{27}}}}\ {\sin{\ }}{3}{x}\ +\ {\frac{{{1}}}{{{125}}}}\ {\sin{\ }}{5}{x}\ +\ {\frac{{{1}}}{{{273}}}}\ {\sin{\ }}{7}{x}\ +\ {\frac{{{1}}}{{{729}}}}\ {\sin{\ }}{9}{x}\right)}\)
image
Step 3
We see that \(\displaystyle{S}_{{{10}}}{\left({x}\right)}\) is a very good a approximation of the function
\(\displaystyle{f{{\left({x}\right)}}}={x}{\left(\pi\ -\ {x}\right)}.\)
The absolute value of their difference is given below for the comparison:
image
0

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