As the title shows,we know that there is some points the series not approaching to the function.


Now,take the convergence theorem into consideration.As there is some the first break-points，the series is still convergent.And,the Gibbs phenomenon always takes place on the first break-points.


Why does Gibbs phenomenon take place?What does it show the nature of Fourier Series?



What does Gibbs phenomenon shows the nature of Fourier Series
A Fourier series truncated to order $n$ is the best approximation to the given function *in the $L^2$ sense* using trigonometric polynomials of order $n$. As such, small rapid deviations don't matter much. Since there is a limit to how big the derivatives of a trigonometric polynomial of fixed order can be (without the coefficients being big), in order to fit such a polynomial to a discontinuity it pays to overshoot a bit on each side of the discontinuity in order to “gather speed” so you can get from one value to the other fast. When I say it “pays”, i mean to say that you what you lose by not approximating the function too well at the overshoot, you more than gain back by doing the jump faster.


