Chapter 14, Fourier Series Video Solutions, Advanced Engineering Mathematics, International Student Edition

Problem 1

On the same set of axes, generate a graph of $x(\pi-x)$ and $\sum_{n=1}^5(4 / \pi)\left(\left[1-(-1)^n\right] / n^3\right) \sin (n x)$ for $0 \leq x \leq \pi$. Repeat this for the partial sums $\sum_{n=1}^{10}(4 / \pi)\left(\left[1-(-1)^n\right] / n^3\right) \sin (n x)$ and $\sum_{n=1}^{20}(4 / \pi)\left(\left[1-(-1)^n\right] / n^3\right) \sin (n x)$. This will give a sense of the correctness of Fourier's intuition in asserting that $x(\pi-x)$ can be accurately represented by $\sum_{n=1}^{\infty}(4 / \pi)\left(\left[1-(-1)^n\right] / n^3\right) \sin (n x)$ on this interval.

Joshua Utley

Numerade Educator

05:36

Problem 2

Prove that a polynomial cannot be a constant multiple of $\sin (n x)$ over $[0, \pi]$, for any positive integer $n$. Hint: One way is to proceed by induction on the degree of the polynomial.

Stark Ledbetter

Numerade Educator

01:26

Problem 3

Prove that a polynomial cannot be equal to a nonzero sum of the form $\sum_{j=0}^n c_j \sin (j x)$ for $0 \leq x \leq \pi$, where the $c_j^{\prime} s$ are real numbers.

Raushan Kumar

Numerade Educator