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 :)