Sum of a serie involving Fourier coefficients

[Mathitalian]

Homework Statement

Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie:

\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2


Ok, I think there is a trick that I don't know...

\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2

becomes

\sum_{k=-\infty}^{\infty}\hat{u}_k

What can i do now? Any help will be appreciated :)

 

[LCKurtz]

Homework Statement

Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie:

\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2


Ok, I think there is a trick that I don't know...

\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2

becomes

\sum_{k=-\infty}^{\infty}\hat{u}_k

What can i do now? Any help will be appreciated :)

Draw the graph of the periodic extension of $f(t) = t$ on [0,2). What happens at $t = 2$? What do you know about the sum of the series there?

 

[Mathitalian]

Well.. u(t) is discontinue for t=2 so the Fourier serie converges to [u(2+)+ u(2-)]/2 where

u(2+)= \lim_{t\to 2^+}u(t)= 0
u(2-)= \lim_{t\to 2^-}u(t)= 2

so

\sum_{k=-\infty}^{\infty}\hat{u}_k= 1

Right?

 

[LCKurtz]

Yes. That's the idea.

 

[Mathitalian]

Thank you so much LCKurtz :)