 Solution

The periodic continuation is $g_{\mathrm{per}}(x)=e^{x-2k\pi }$ on each interval $-\pi +2k\pi \le x<\pi +2k\pi$. Thus the plot repeats an increasing exponential segment on each of the three requested periods, jumping down from $e^{\pi }$ to $e^{-\pi }$ at odd multiples of $\pi$.

At $x=\pi$, the left and right limits are $e^{\pi }$ and $e^{-\pi }$. The Dirichlet-Jordan convergence theorem therefore makes the Fourier series converge toThe actual half-open periodic definition has $g_{\mathrm{per}}(\pi )=e^{-\pi }$; the Fourier value at the jump need not equal that assigned endpoint value.