Upper bound for the error on the Fourier series for $e^{x}$

Question

I have been given the following problem:

Find the Fourier series for $e^{x}$ over the interval $-\pi \le t \le \pi$. Hence find the upper bound of its error.

To spare me typing a huge expression and to spare you the reading, I will simply link to the Wolfram Alpha result: http://www.wolframalpha.com/input/?i=fourier+series+exp%28x%29

Unfortunately Wolfram Alpha could not compute over intervals, so I used MATLAB to plot the 4th order Fourier series over the interval, and by investigation I saw that the largest error value must be at $t = \pi$. I assumed then that the upper error bound must be the value of the fourier series at $t = \pi$ minus the value of $e^{\pi}$ (or the absolute value of this). I got something ~13.6.

Is this the correct way of thinking about this? Also, does the same value hold as the order of the fourier series approaches infinity (i.e number of sums gets larger and larger?).

Answer 1

The sudden jump at $t = \pi$ is $e^\pi - e^{-\pi} \approx 23.10$.

Gibbs phenomenon occurs at this discontinuity, and the overshoot is approximately $23.10 \cdot 0.0894 \approx 2.07$