Free Printable Fourier Series

1.

Find the $a_0$ of Fourier Series of f(x)= $x^2,\ \ \ 0<x<2\pi$

a)

$\frac{4\pi^2}{3}$

b)

$\frac{2\pi^2}{3}$

c)

$-\frac{4\pi^2}{3}$

d)

$\frac{4\pi^{ }}{3}$

2.

Find the $a_n$ of Fourier Series of f(x)= $x^2,\ \ \ 0<x<2\pi$

a)

$\frac{4\pi}{n}$

b)

$\frac{4}{n}$

c)

$-\frac{4\pi}{n}$

d)

$\frac{2\pi}{n}$

3.

Find the $a_n$ of Fourier Series of f(x)= $e^{-x},\ \ \ 0<x<2\pi$

a)

$-\frac{n\left(1-e^{-2\pi}\right)}{\pi\left(1+n^2\right)}$

b)

$\frac{n\left(1-e^{-2\pi}\right)}{\pi\left(1+n^2\right)}$

c)

$\frac{\left(1-e^{-2\pi}\right)}{\pi\left(1+n^2\right)}$

d)

$\frac{n\left(1-e^{-2\pi}\right)}{\left(1+n^2\right)}$

4.

Find the $b_n$ of Fourier Series of f(x)= $e^x,\ \ \ -\pi<x<\pi$

a)

$-\frac{2n\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

b)

$\frac{2n\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

c)

$-\frac{2\cos nx}{\pi\left(1+n^2\right)}.\sinh\pi$

d)

$-\frac{2n\cos nx}{\left(1+n^2\right)}.\sinh\pi$

5.

Find the $b_n$ of Fourier Series of f(x)= $x^2,\ \ \ -\pi<x<\pi$

a)

Zero

b)

$\frac{3\pi}{2}$

c)

$-2\pi$

d)

$-1$

6.

Find the $a_0$ of Fourier Series of f(x)= $4-x^2,\ \ \ 0<x<2$

a)

$\frac{8}{3}$

b)

$\frac{7}{3}$

c)

$\frac{4}{3}$

d)

$\frac{5}{3}$

7.

Fond the fourier series of $f(x)=x\left|x\right|\ \ \ \ in\ \ \ (-1,1)$

a)

$-2\sum_{ }^{ }\left[-\frac{\left(-1\right)^n}{n\pi}+\frac{2\left(-1\right)^n}{n^3\pi^3}-\frac{2}{n^3\pi^3}\right]\sin n\pi x$

b)

$2\sum_{ }^{ }\left[-\frac{\left(-1\right)^n}{n\pi}+\frac{2\left(-1\right)^n}{n^3\pi^3}-\frac{2}{n^3\pi^3}\right]\sin n\pi x$

c)

$2\sum_{ }^{ }\left[\frac{\left(-1\right)^n}{n\pi}+\frac{2\left(-1\right)^n}{n^3\pi^3}-\frac{2}{n^3\pi^3}\right]\sin n\pi x$

d)

$2\sum_{ }^{ }\left[-\frac{\left(-1\right)^n}{n\pi}+\frac{2\left(-1\right)^n}{n^3\pi^3}+\frac{2}{n^3\pi^3}\right]\sin n\pi x$

8.

Find the F.S. of $f\left(x\right)\ =\ -\cos x\ \ \ \ in\ \ -\pi\le x\le\pi$

a)

$-\frac{2n\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

b)

$\frac{2n\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

c)

$\frac{2n\left(1+\cos n\pi\right)}{\left(n^2-1\right)}\ \ if\ \ n\ne1$

d)

$\frac{2\left(1+\cos n\pi\right)}{\pi\left(n^2-1\right)}\ \ if\ \ n\ne1$

9.

Find the $a_0$ F.S of $f\left(x\right)\ =\ 2x-x^2\ \ ,\ 0\le x\le3$

a)

$Zero$

b)

$2\pi$

c)

$-\pi$

d)

$1$

10.

Obtain the fourier expansion of $x-x^3$ from $(-1,1)$

a)

$f\left(x\right)=-\frac{12}{\pi^3}\sum_{n=1}^{\infty}\frac{\left(-1\right)^n\sin n\pi x}{n^3}$

b)

$f\left(x\right)=\frac{12}{\pi^3}\sum_{n=1}^{\infty}\frac{\left(-1\right)^n\sin n\pi x}{n^3}$

c)

$f\left(x\right)=-\frac{12}{\pi^3}\sum_{n=1}^{\infty}\frac{\left(-1\right)^n\sin n\pi x}{n^{ }}$

d)

$f\left(x\right)=-\frac{12}{\pi^{ }}\sum_{n=1}^{\infty}\frac{\left(-1\right)^n\sin n\pi x}{n^3}$

Fourier Series

Total questions: 10

Worksheet time: 50mins

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