[tex]f(x)=\begin{cases}1&\text{for }-7<x<0\\1+x&\text{for }0\le x<7\end{cases}[/tex]
The Fourier series expansion of [tex]f(x)[/tex] is given by
[tex]\dfrac{a_0}2+\displaystyle\sum_{n\ge1}a_n\cos\frac{n\pi x}7+\sum_{n\ge1}b_n\sin\frac{n\pi x}7[/tex]
where we have
[tex]a_0=\displaystyle\frac17\int_{-7}^7f(x)\,\mathrm dx[/tex]
[tex]a_0=\displaystyle\frac17\left(\int_{-7}^0\mathrm dx+\int_0^7(1+x)\,\mathrm dx\right)[/tex]
[tex]a_0=\dfrac{7+\frac{63}2}7=\dfrac{11}2[/tex]
The coefficients of the cosine series are
[tex]a_n=\displaystyle\frac17\int_{-7}^7f(x)\cos\dfrac{n\pi x}7\,\mathrm dx[/tex]
[tex]a_n=\displaystyle\frac17\left(\int_{-7}^0\cos\frac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\cos\frac{n\pi x}7\,\mathrm dx\right)[/tex]
[tex]a_n=\dfrac{9\sin n\pi}{n\pi}+\dfrac{7\cos n\pi-7}{n^2\pi^2}[/tex]
[tex]a_n=\dfrac{7(-1)^n-7}{n^2\pi^2}[/tex]
When [tex]n[/tex] is even, the numerator vanishes, so we consider odd [tex]n[/tex], i.e. [tex]n=2k-1[/tex] for [tex]k\in\mathbb N[/tex], leaving us with
[tex]a_n=a_{2k-1}=\dfrac{7(-1)-7}{(2k-1)^2\pi^2}=-\dfrac{14}{(2k-1)^2\pi^2}[/tex]
Meanwhile, the coefficients of the sine series are given by
[tex]b_n=\displaystyle\frac17\int_{-7}^7f(x)\sin\dfrac{n\pi x}7\,\mathrm dx[/tex]
[tex]b_n=\displaystyle\frac17\left(\int_{-7}^0\sin\dfrac{n\pi x}7\,\mathrm dx+\int_0^7(1+x)\sin\dfrac{n\pi x}7\,\mathrm dx\right)[/tex]
[tex]b_n=-\dfrac{7\cos n\pi}{n\pi}+\dfrac{7\sin n\pi}{n^2\pi^2}[/tex]
[tex]b_n=\dfrac{7(-1)^{n+1}}{n\pi}[/tex]
So the Fourier series expansion for [tex]f(x)[/tex] is
[tex]f(x)\sim\dfrac{11}4-\dfrac{14}{\pi^2}\displaystyle\sum_{n\ge1}\frac1{(2n-1)^2}\cos\frac{(2n-1)\pi x}7+\frac7\pi\sum_{n\ge1}\frac{(-1)^{n+1}}n\sin\frac{n\pi x}7[/tex]
