Compute
And if you cannot find it see how closely you can bound it.
Let be function such that . Furthermore let have the charcteristic that . Now it is pretty clear that except possibly at , but in fact is continuous at all these points due to its eveness. Next note that , in other words its posses a right and left hand limit everywhere. So since noting all of these things we may conclude that where is the Fourier Series given by.
Where
And as was stated above
So
Now letting gives
So now consider when we are left with
Solving gives
So then we get finally that
Note from this we can also glean that
Hey! Don't be a Negative Nancy! Haha, your right though
I like this solution because it is simplistic and powerful and in so beautiful. I dislike it because it doesn't take much thought. Picking a Fourier Series is much harder than one would imagine. Also, it can become repetitive, not that my Fourier Series are that much different each time though...and FS looks cooler Regardless, thanks again Paul for a nice solution.