# Write seven terms of the Fourier series given the following coefficients?

## a0=4, a1=3, a2=2,a3=1;b1=7.5, b2=5.3, b3=2.8

Jan 5, 2018

$2 + 3 \cos \left(\frac{\pi x}{l}\right) + 2 \cos \left(\frac{2 \pi x}{l}\right) + 1 \cos \left(\frac{3 \pi x}{l}\right) + 7.5 \sin \left(\frac{\pi x}{l}\right) + 5.3 \sin \left(\frac{2 \pi x}{l}\right) + 2.8 \sin \left(\frac{3 \pi x}{l}\right) + \ldots$

#### Explanation:

For a periodic function $f \left(x\right)$ where the period is $l$ the Fourier expansion is given as:

$f \left(x\right) = {a}_{0} / 2 + {\sum}_{n = 1}^{\infty} {a}_{n} \cos \left(\frac{n \pi x}{l}\right) + {b}_{n} \sin \left(\frac{n \pi x}{l}\right)$

So substituting the the given coefficients into the fourier expansion the first 7 terms will be:

$f \left(x\right) = {a}_{0} / 2 + {a}_{1} \cos \left(\frac{\pi x}{l}\right) + {a}_{2} \cos \left(\frac{2 \pi x}{l}\right) + {a}_{3} \cos \left(\frac{3 \pi x}{l}\right) + {b}_{1} \sin \left(\frac{\pi x}{l}\right) + {b}_{2} \sin \left(\frac{2 \pi x}{l}\right) + {b}_{3} \sin \left(\frac{3 \pi x}{l}\right) + \ldots$

$= 2 + 3 \cos \left(\frac{\pi x}{l}\right) + 2 \cos \left(\frac{2 \pi x}{l}\right) + 1 \cos \left(\frac{3 \pi x}{l}\right) + 7.5 \sin \left(\frac{\pi x}{l}\right) + 5.3 \sin \left(\frac{2 \pi x}{l}\right) + 2.8 \sin \left(\frac{3 \pi x}{l}\right) + \ldots$