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].

Question

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].

Layton · Accepted Answer

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

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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