# Partial sum of Fourier series

I found the coefficients of the Fourier series. The partial sum of the Fourier series is defined on the interval $$[0,2\pi]$$ by

$$S_N(x) = \frac{a_0}{2} + \sum_{n=1}^{N}[ a_n \cos(nx) + b_n \sin(nx)],$$

where

$$a_0 =\frac{1}{\pi} \int_{0}^{2\pi} f(x)\, dx,$$ $$a_n = \frac{1}{\pi} \int_{0}^{2\pi} f(x) \cos(nx)\, dx,$$ $$b_n =\frac{1}{\pi} \int_{0}^{2\pi} f(x) \sin(nx)\, dx.$$

Is there any way to put all the coefficients in the partial sum and to derive the function? I struggled after I put everything in one equation to see if it can be simplified more.

• I am new to this website, I try to fix the way to write equations. I am so sorry Dec 29, 2021 at 9:04
• Have a look at this.
– Gary
Dec 29, 2021 at 9:10
• Thank you ! @Gary Dec 29, 2021 at 9:56
• At the moment it is not clear what you are asking for. What would you like to achive/do exactly?
– Gary
Dec 29, 2021 at 10:05
• if I plug a0,an,and bn into S_N(x). How could be simplified more ? Can we solve the sum or S_n(x) be written in simple way ? Dec 29, 2021 at 15:42