= Solution
The periodic continuation is $g_{\rm per}(x)=e^{x-2k\pi}$ on each interval $-\pi+2k\pi\leq x<\pi+2k\pi$. 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 to
$$
\boxed{\frac{e^\pi+e^{-\pi}}2=\cosh\pi.}
$$
The actual half-open periodic definition has $g_{\rm per}(\pi)=e^{-\pi}$; the Fourier value at the jump need not equal that assigned endpoint value.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2014/paper-1-periodic-exponential.png]
{title=Three periods of the exponential continuation and the Fourier-series midpoint at a jump}
Back to article page
