Can the Fourier Series for This Step Function Be Determined by Observation?

[neelakash]

I want to solve this by observation:

Consider a step function:

F(t)=1 for 0<t< $$\frac{\pi}{\omega}$$

F(t)=-1 for -$$\frac{\pi}{\omega}$$<t<0

and the same pattern is repeated over time.

What should be the Fourier Series:

(A) $$\frac{4}{\pi}$$ $$\sum^\infty_1\frac{sin(n\omega\ t )}{n}$$


(B) $$\frac{4}{\pi}$$ $$\sum^\infty_0\frac{sin(2n+1)\omega\ t }{2n+1}$$



It is easy to check that the actual answer is (B);but I want to see if this can be seen by observation only...(A) and (B) differe in that (A) contains the even terms as well where (B) lacks them...

Can anyone say something if it could be done by using observation?

 

[neelakash]

OK,realize that the given function is symmetric about
$$\ t=$$$$\frac{\pi}{2\omega}$$

Then,try to sketch the graphs of even terms of sine (like n=2,n=4...) adjusting the scale on the t axis.(A) will show zeros at $$\frac{\pi}{2\omega}$$