Fourier series

Find the Fourier series of the $2\pi$-periodic extension of

$$f(x)=\{\begin{array}{ll}-2& \text{if }-\pi \le x<0\\ 2& \text{if }0\le x<\pi \end{array}$$

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Find the Fourier series of the $2\pi$-periodic extension of

$$f(x)=\{\begin{array}{ll}0& \text{if }-\pi \le x<0\\ 2& \text{if }0\le x<\pi \end{array}$$

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Find the Fourier series of the $2\pi$-periodic extension of $f(x)=\pi -|x|$, $x\in [-\pi ,\pi ]$.

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Find the Fourier series of $f(x)={\sin }^{3}x$ without computing any integral.

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Find the Fourier series of the $2\pi$-periodic extension of $f(x)=\frac{x}{2}$, $x\in [-\pi ,\pi )$.

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Find the Fourier series of the $2\pi$-periodic extension of $f(x)=(x-1{)}^2$, $x\in [-\pi ,\pi )$.

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Let $f$ be the $2\pi$-periodic extension of

$$f(x)=\{\begin{array}{ll}{x}^2& \text{if }-\pi \le x<0\\ 1& \text{if }0\le x<\pi \end{array}$$

Find, without computing the coefficients, the sum of the Fourier series of $f$ at $x=0$ and at $x=\pi$.

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Let $f$ be the $2\pi$-periodic extension of $f(x)=e^x$, $x\in [-\pi ,\pi )$. Find, without computing the coefficients, the sum of the Fourier series of $f$ at $x=0$ and at $x=\pi$.

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Find the Fourier series of the $2\pi$-periodic extension of

$$f(x)=\{\begin{array}{ll}-1& \text{if }-\pi \le x<0\\ 1& \text{if }0\le x<\pi \end{array}$$

and, by evaluating it at $x=\frac{\pi }{2}$, compute the sum of the numerical series $\sum _{k=0}^{\infty }\frac{(-1{)}^k}{2k+1}$.

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Given that the $2\pi$-periodic extension of $f(x)=x^3-{\pi }^{2}x$, $x\in [-\pi ,\pi ]$, has Fourier series

$$x^3-{\pi }^{2}x\sim \sum _{n=1}^{\infty }\frac{12(-1{)}^n}{n^3}\sin (nx),$$

compute with Parseval’s identity the sum of the series $\sum _{n=1}^{\infty }\frac{1}{n^6}$.

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Find the Fourier series of the $4$-periodic extension of $f(x)=x$, $x\in [-2,2)$.

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Find the Fourier series of the $2$-periodic extension of $f(x)=|x|$, $x\in [-1,1]$, and determine whether the convergence is uniform on $\mathbb{R}$.

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