Gaussian Integral: Infinite Series Approach

We evaluate the Gaussian integral, using an approach based on series expansions.

This argument can be traced back to a Stack Exchange post here:

Exchanging the order of summation and integration:

00:00 Setup
02:05 Infinite series
04:05 Integrating
08:03 Partial fractions
10:59 Simplifying the inner sum
14:15 Back to integration
17:45 Back to powers of e
21:23 Taking limits

[blackpenredpen]

WOW!!! I never knew this was possible with infinite series!

[umylten]

That's pretty funny actually, one of the other ways to do it is using Feynman's trick on I(t) = int(-inf, inf) exp(-t²(x²+1))/(x²+1)dx, and that integrand appears at the very end of your calculations.

In a sense the series expansion method explains why it's natural to use that function for Feynman's trick.

[bjornfeuerbacher]

More complicated than the usual approach, but on the other hand, one does not need to know polar coordinates... Very nice way! :) Did you come up with that on your own?

[peterhall]

I hadn't seen this one before but it is straightforward one you break it down as you have done. Paul Levy's original functional analysis treatise which Norbert Wiener used for his development of Brownian motion actually developed the Gaussian from volumes of n dimensional spheres. As usual you have done a very clear and succinct exposition.

[mathijsj]

Ok, ok, you convinced me. I'll learn polar coordinates.

Nice video! :)

[Homayoun]

Thank you so much for this interesting approach other than the polar coordinates one.

[VeteranVandal]

There are other methods that are shorter, but this one certainly does the job.

[radadadadee]

I thought it was overcomplicated until the mid part where I really liked how many little surprises it had! I enjoyed it. The ending with the "ok, and this messy part will go to zero" was kinda underwhelming tbh, but good video overall.

[__a_]

Out of interest, what prevents applying the power series method without squaring the integral?

Edit: I'm going to try it out and see where I hit issues

[WielkiKaleson]

Always squaring... not that I am complaining, but it is - I guess - a highly surprising trick.