Fourier series Basics

1

0

-1

n

0 and 0

 $\left(-1\right)^n$   and 0

 $\left(-1\right)^n$  and 1

1 and 1

0 and 0

1 and 1

1 and -1

-1 and -1

5 sin5x

-5sin5x

 $\frac{\sin5x}{5}$  

 $-\frac{\sin5x}{5}$  

 $\frac{2}{3}$  

 $-\frac{2}{3}$  

0

 $\frac{1}{3}$  

x

2

2x

0

1

k

0

x

 $\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|$  

 $\left|x\left(\cos x\right)-\left(\sin x\right)\right|$  

 $\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ 0$  

 $\left|x\left(-\cos x\right)-\left(-\sin x\right)\right|_0^{2\pi}\ =\ -2\pi$  

 $\left|x\left(\sin x\right)\left(-\cos x\right)\right|$  

 $\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|$  

 $\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ \ =0$  

 $\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\ =2$  

 $\left|x\left(\sin\left(\frac{n\pi}{l}\ \right)x\right)\left(-\cos\left(\frac{n\pi}{l}\ \right)x\right)\right|$  

 $\left|x\left(\frac{\sin\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)}\right)-\left(-\frac{\cos\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)^2}\right)\right|$  

 $\left|x\left(\frac{\sin\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)}\right)-\left(-\frac{\cos\left(\frac{n\pi}{l}\ \right)x}{\left(\frac{n\pi}{l}\ \right)^2}\right)\right|_0^{2\pi}\ \$                           

    = 0

 $\left|x\left(\frac{\sin nx}{n}\right)-\left(-\frac{\cos nx}{n^2}\right)\right|_0^{2\pi}\$  
 =2