be taking the limit as n goes to infinity when i take a large value of n what the e to the minus n times this does is it just looks like this very narrow region that's zero pretty much everywhere else and recall that we did this taylor expansion around z equal to one which was the maximum of logarithm of z minus z if you had chosen to expand around some other point the scenario would be even worse you'd have e to an even more negative of a number and the collapsing away from this approximation point would go down to zero even faster now i acknowledge i'm being a little bit hand wavy here and saying okay it's going to go into zero very quickly it's not going to matter there is a formal proof that needs to be done this is called laplace's method and for a generic category of such f which logarithm of z minus z satisfies this will always work out but i'll put a link down in the description so you can see the larger proof of this method if you're so interested regardless what i've tried to argue is that a taylor approximation which is only valid around the point is reasonable due because away from that point is dropping off to zero and it's not going to matter to the integral anyways the gaussian integral is going to give the value of root pi but because there is an n over NUMBER_TOKEN on the top it becomes root pie NUMBER_TOKEN divided by n instead i'm gonna put the n to the n and the e to the minus n together i'm going to factor out the one extra n and i get my final version with a little bit of algebra stated the way sterling's formula is and so we finally have it n factorial is approximately equal to n divided by e to the power of n square root of two pi n we're done my thanks once again to maplesoft for sponsoring this video i definitely encourage you to check out either maple calculator on the app store or the play store which at least in my opinion is the best and most powerful free calculator app or maple learn for the browser which allows you to make these wonderful mathematics documents where they do the computations for you they do the graphics for you it's really cool so check out those maple products other than that if you have any questions about this video leave them down in the comments below and we'll do some more math in the next video
START_SPONSOR_TOKEN my thanks once again to maplesoft for sponsoring this video i definitely encourage you to check out either maple calculator on the app store or the play store which at least in my opinion is the best and most powerful free calculator app or maple learn for the browser which allows you to make these wonderful mathematics documents where they do the computations for you they do the graphics for you it's really cool so check out those maple products END_SPONSOR_TOKEN