write-the-fourier-series-of-f-x-x-0-x-2-

Question Number 50702 by pooja24 last updated on 19/Dec/18

w r i t e t h e f o u r i e r s e r i e s o f

f (x) = x 0 ⩽ x ⩽ 2

Commented by maxmathsup by imad last updated on 19/Dec/18

i t h i n k 0 ⩽ x ⩽ 2 π i f f i s 2 π p e r i o d i c o d d w e h a v e

f (x) = \sum_{n = 1}^{\infty} a_{n} s i n (n x) w i t h a_{n} = \frac{2}{T} \int_{[T]} f (x) s i n (n x) d x

= \frac{2}{2 π} \int_{- π}^{π} x s i n (n x) d x = \frac{2}{π} \int_{0}^{π} x s i n (n x d x \Rightarrow \frac{π}{2} a_{n} = \int_{0}^{π} x s i n (n x) d x

b y p a r t s \int_{0}^{π} x s i n (n x) d x = {[- \frac{x}{n} c o s (n x)]}_{0}^{π} + \int_{0}^{π} \frac{1}{n} c o s (n x) d x

= - \frac{π}{n} {(- 1)}^{n} + \frac{1}{n} {[\frac{1}{n} s i n (n x)]}_{0}^{π} = \frac{π {(- 1)}^{n - 1}}{n} \Rightarrow \frac{π}{2} a_{n} = \frac{π}{n} {(- 1)}^{n - 1} \Rightarrow

a_{n} = \frac{2}{n} {(- 1)}^{n - 1} \Rightarrow f (x) = \sum_{n = 1}^{\infty} \frac{2}{n} {(- 1)}^{n - 1} s i n (n x) .