Question 12. Consider the function ø(x) = e + x. 2 12a. W down the Fourier series and the Fourier cosine series of ø(x) on the interval x E (0, 1) but leave your coefficients as integrals. 12b. Write down the full Fourier series of ø(x) on the interval x E (-1,1) but leave your coefficients as integrals. 12c. At the point x = 0, what number does the Fourier sine series in 8a converge to? In other words, if we take more and more terms in the sum and evaluate at x = 0, what number will we get close to? What number does the Fourier cosine series in 12a converge to? What number does the full Fourier series in 12b converge to? 12d. For each series, plot the function that the Fourier series will get closer to (converge to) on the interval x E (-2, 2).

Given function is

$\varphi \left(x\right)=e^x+x$.

Solution of 12a.

Fourier sine series of $\varphi \left(x\right)$ on the interval $\left(0,1\right)$ is

$\varphi \left(x\right)=\sum _{n=1}^{\infty }{b}_n\sin \left(n\mathrm{\pi x}\right)$

or

$e^x+x=\sum _{n=1}^{\infty }{b}_n\sin \left(n\mathrm{\pi x}\right)$

where 

$b_n=2{\int }_0^1\left(e^x+x\right)\sin \left(n\mathrm{\pi x}\right)dx$.

Fourier cosine series of $\varphi \left(x\right)$ on the interval $\left(0,1\right)$ is

$\varphi \left(x\right)=\frac{a_0}{2}+\sum _{n=1}^{\infty }{a}_{n}cos\left(n\mathrm{\pi x}\right)$

or 

$e^x+x=\frac{a_0}{2}+\sum _{n=1}^{\infty }{a}_{n}cos\left(n\mathrm{\pi x}\right)$

where 

$$\begin{gathered} a_0=2{\int }_0^1\left(e^x+x\right)dx \\ and \\ {a}_n=2{\int }_0^1\left(e^x+x\right)cos\left(n\mathrm{\pi x}\right)dx \end{gathered}$$