Recently i have been reading "A radical approach to real analysis" by David Bressoud and this thing got stuck in my mind 
Here you might lose convergence at the end point is written in last paragraph what is this meant for an infinite trigonometric series .It has also some relation with derivatives somehow. This was written for the explanation of fourier series and crisis in mathematics but what means by phrase losing convergence and how it is related to trigonometric infinite series
I think Bressoud is saying that when you switch from finite to infinite sums of powers the worst that can happen is loss of convergence at endpoints. To see how much nastier things can get when you allow infinite sums of sines and cosines read about pointwise convergence of Fourier series.