Question

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`

Answer 1

`2+3cos((pix)/l)+2cos((2pix)/l)+1cos((3pix)/l)+7.5sin((pix)/l)+5.3sin((2pix)/l)+2.8sin((3pix)/l)+...`

Explanation:

For a periodic function `f(x)` where the period is `l` the Fourier expansion is given as:

`f(x) =a_0/2 + sum_(n=1)^ooa_ncos((n pi x)/l)+b_nsin((n pi x)/l)`

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

`f(x) =a_0/2+a_1cos((pix)/l)+a_2cos((2pix)/l)+a_3cos((3pix)/l)+b_1sin((pix)/l)+b_2sin((2pix)/l)+b_3sin((3pix)/l)+...`

`=2+3cos((pix)/l)+2cos((2pix)/l)+1cos((3pix)/l)+7.5sin((pix)/l)+5.3sin((2pix)/l)+2.8sin((3pix)/l)+...`