Mathematics-I (3110014)

$\mathrm{Here},\mathrm{f}\left(\mathrm{x}\right)={\mathrm{e}}^{\mathrm{x}}$

$\mathrm{Also}, (0,\mathrm{L})= (0,\mathrm{l})$

$⟹\mathrm{L}=\mathrm{l}$

${\mathrm{a}}_0=\frac{2}{\mathrm{L}}{\int }_0^{\mathrm{L}}\mathrm{f}\left(\mathrm{x}\right) d\mathrm{x}$

${\mathrm{a}}_0=\frac{2}{\mathrm{l}}{\int }_0^{\mathrm{l}} {\mathrm{e}}^{\mathrm{x}} d\mathrm{x}$

${\mathrm{a}}_0=\frac{2}{\mathrm{l}}{\left[{\mathrm{e}}^{\mathrm{x}}\right]}_0^{\mathrm{l}}$

${\mathrm{a}}_0=\frac{2}{\mathrm{l}}\left[{\mathrm{e}}^{\mathrm{l}}-{\mathrm{e}}^0\right]$

${\mathrm{a}}_0=\frac{2}{\mathrm{l}}\left[{\mathrm{e}}^{\mathrm{l}}-1\right]$

${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\mathrm{L}}{\int }_0^{\mathrm{L}}\mathrm{f}\left(\mathrm{x}\right) \cos \left(\frac{\mathrm{n\pi x}}{\mathrm{L}}\right) d\mathrm{x}$

${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\mathrm{l}}{\int }_0^{\mathrm{l}}{\mathrm{e}}^{\mathrm{x}} \cos \left(\frac{\mathrm{n\pi x}}{\mathrm{l}}\right) d\mathrm{x} \left[\mathrm{Here}, \mathrm{a}=1 ; \mathrm{b}=\frac{\mathrm{n\pi }}{\mathrm{l}}\right]$

${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\mathrm{l}}{\left[\frac{{\mathrm{e}}^{\mathrm{x}}}{1+{\displaystyle \frac{{\mathrm{n}}^2 {\mathrm{\pi }}^2}{{\mathrm{l}}^2}}}\left\{\left(1\right)\cos \left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\sin \left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\right\}\right]}_0^{\mathrm{l}}$

${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\mathrm{l}}\left(\frac{{\mathrm{l}}^2}{{\mathrm{l}}^2+{\mathrm{n}}^2 {\mathrm{\pi }}^2}\right){\left[{\mathrm{e}}^{\mathrm{x}}\left\{\left(1\right)\cos \left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\sin \left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\right\}\right]}_0^{\mathrm{l}}$

${\mathrm{a}}_{\mathrm{n}}=\frac{2}{\mathrm{l}}\left(\frac{{\mathrm{l}}^2}{{\mathrm{l}}^2+{\mathrm{n}}^2 {\mathrm{\pi }}^2}\right)\left[{\mathrm{e}}^{\mathrm{l}}\left\{\cos \left(\mathrm{n\pi }\right)+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\sin \left(\mathrm{n\pi }\right)\right\}-{\mathrm{e}}^0\left\{\cos \left(0\right)+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\sin \left(0\right)\right\}\right]$

${\mathrm{a}}_{\mathrm{n}}=\frac{2\mathrm{l}}{{\mathrm{l}}^2+{\mathrm{n}}^2 {\mathrm{\pi }}^2}\left[{\mathrm{e}}^{\mathrm{l}}\left\{{\left(-1\right)}^{\mathrm{n}}+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\left(0\right)\right\}-{\mathrm{e}}^0\left\{1+\left(\frac{\mathrm{n\pi }}{\mathrm{l}}\right)\left(0\right)\right\}\right]$

${\mathrm{a}}_{\mathrm{n}}=\frac{2\mathrm{l}}{{\mathrm{l}}^2+{\mathrm{n}}^2 {\mathrm{\pi }}^2}\left[{\mathrm{e}}^{\mathrm{l}}{\left(-1\right)}^{\mathrm{n}}-1\right]$

$\mathrm{Now}, \mathrm{Fourier} \mathrm{series} \mathrm{of} \mathrm{f}\left(\mathrm{x}\right) \mathrm{is},$

$\mathrm{f}\left(\mathrm{x}\right)=\frac{{\mathrm{a}}_0}{2}+\sum _{\mathrm{n}=1}^{\infty }{\mathrm{a}}_{\mathrm{n}} \cos \left(\frac{\mathrm{n\pi x}}{\mathrm{l}}\right)$

$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{1}{\mathrm{l}}\left[{\mathrm{e}}^{\mathrm{l}}-1\right]+\sum _{\mathrm{n}=1}^{\infty }\left[\frac{2\mathrm{l}}{{\mathrm{l}}^2+{\mathrm{n}}^2 {\mathrm{\pi }}^2}\left({\mathrm{e}}^{\mathrm{l}}{\left(-1\right)}^{\mathrm{n}}-1\right)\right]\cos \left(\frac{\mathrm{n\pi x}}{\mathrm{l}}\right)$