Question

5. [4+8+3 marks] Consider the function $f(x) = \begin{cases} 0 & \text{if } -\frac{1}{2} \le x < 0\\ 1 & \text{if } 0 \le x < \frac{1}{2} \\ \pi \sin \pi x & \text{if } \frac{1}{2} < x \le \frac{1}{2} \end{cases}$ (a) Use the convergence theorem to find the sum of the Fourier series of $f$ on $[-\frac{1}{2}, \frac{1}{2}]$. (Do not find the Fourier series). (b) Find the Fourier cosine series of the function $g(x) = \pi \sin \pi x$ on $0 \le x \le \frac{1}{2}$. (c) Use your answer in (b) to show that $\sum_{n=1}^{\infty} \frac{1}{(2n - 1)(2n + 1)} = \frac{1}{2}$.

          5. [4+8+3 marks] Consider the function $f(x) = \begin{cases} 0 & \text{if } -\frac{1}{2} \le x < 0\\ 1 & \text{if } 0 \le x < \frac{1}{2} \\ \pi \sin \pi x & \text{if } \frac{1}{2} < x \le \frac{1}{2} \end{cases}$
(a) Use the convergence theorem to find the sum of the Fourier series of $f$ on $[-\frac{1}{2}, \frac{1}{2}]$. (Do not find the Fourier series).
(b) Find the Fourier cosine series of the function $g(x) = \pi \sin \pi x$ on $0 \le x \le \frac{1}{2}$.
(c) Use your answer in (b) to show that $\sum_{n=1}^{\infty} \frac{1}{(2n - 1)(2n + 1)} = \frac{1}{2}$.

5. [4+8+3 marks] Consider the function f(x) = 0 if -(1)/(2)≤ x < 0
1 if 0 ≤ x < (1)/(2)
πsinπ x if (1)/(2) < x ≤(1)/(2)
(a) Use the convergence theorem to find the sum of the Fourier series of f on [-(1)/(2), (1)/(2)]. (Do not find the Fourier series).
(b) Find the Fourier cosine series of the function g(x) = πsinπ x on 0 ≤ x ≤(1)/(2).
(c) Use your answer in (b) to show that ∑n=1^∞(1)/((2n - 1)(2n + 1)) = (1)/(2).

Added by David M.

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