Some cute integrals

Normally I would hate evaluating integrals with trig functions with the white hot intensity of a thousand suns, but these are so cute they can get away with it.



Everybody get  . Good!
Next one



Everybody get  . Good!
Next one



Keep doing these all the way until



Big surprise!  What did you get?

Before doing the next one, guess the next one.

If you guessed   , you would be wrong. It is actually about 2.31e-11 lower than that. Still a rational multiple of   though. If you managed to guess that I tip my hat to you.

You can't possibly be satisfied with this. Something has to explain why the jump suddenly happened so far into the sequence.

Well, to do these better double the integral by going over the entire real line. Now use the property that integrating in real space and in momentum space gives the same results. The products of sincs becomes convolutions of box functions. In fact you might as well do this with a general sequence like so:

 instead of the reciprocals of the odd integers like we did before.

Look at the running sums of the alphas we had before. You see that



The step where things mess up is when the sum jumps above 2. This is generally true as well. To show why this is true you need to look at the widths of the convolutions and how that grows.

Next time someone tells you to fill in the pattern



you can confidently tell them you have no fucking clue.