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

DofotheroU 2021-01-05 Answered
Let SN(x)=4π n=1N 1  (1)nn3 sin(nx).
Construct graphs of SN(x) and x(π  x), for 0  x  π, 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
4π n=1 1  (1)nn3 sin(nx) on [0, π].
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Expert Answer

Layton
Answered 2021-01-06 Author has 89 answers

Step 1
Note that for even n the coefficient in the sum is zero, so some terms are absent.
image
Step 2
S10(x)=8π(sin x + 127 sin 3x + 1125 sin 5x + 1273 sin 7x + 1729 sin 9x)
image
Step 3
We see that S10(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:
image

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