
Complex fourier series
While deriving complex fourier series I have reached at $\displaystyle c_0+\sum_{n=1}^{n=\infty}c_ne^{\frac{in\pi x}{l}}+\sum_{n=1}^{n=\infty}c_ne^\frac{in\pi x}{l}$.
Can anyone tell me what combined expression (pattern) can be written for this and why?

You notice that when $\displaystyle n = 0$ then $\displaystyle c_n e^{\frac{i n \pi x}{L}}= c_0 $ so your entire sum can be written as
so $\displaystyle \sum \limits_{n = \infty}^{1} c_n e^{\frac{i n \pi x}{L}} + c_0 + \sum \limits_{n = 1}^{\infty} c_n e^{\frac{i n \pi x}{L}} = \sum \limits_{n = \infty}^{\infty} c_n e^{\frac{i n \pi x}{L}} $