e=(1+1/n)^n Imaginary Interest Rates an e^-pi=-1 Identity

Imaginary interest rates | Ep. 5 Lockdown live math

In this video this relation is used

e = ( 1 + 1/n )^n as n goes to infinity

and

e^x = ( 1 + x/n )^n

you can see easily by expanding the right hand side that

e^x = ( 1 + x/n )^n = 1 + x/(1!) +x^2/(2!) + x^3 / (6!) + ....

From this relations you can easily see that you reach the complex number e^ix by a sequence of infinitesimal angular steps with in this case x being the angle on the unit circle.

An interesting thing about e^x = ( 1 + x/n )^n = 1 + x/(1!) +x^2/(2!) + x^3 / (6!) + ....

is that you can see the coefficients by the following method:

to expand ( 1 + x/n )^n = 1 + x/(1!) +x^2/(2!) + x^3 / (6!) + ....

degree zero term: 1 * 1 * 1 * 1 ….*1 with n ones
degree one term with n positions of x meaning the result is
degree two term with n-1 positions of x meaning the result is

With the n factorial term coming from the multiple count of the same positions.

Now due to the double counting of the terms in the 2nd degree tally you need to divide by 2!. In fact for each degree you must divide by n! to back out the multiple counting.