Imaginary interest rates | Ep. 5 Lockdown live math

In this video this relation is used

as n goes to infinity

and

you can see easily by expanding the right hand side that

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

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

to expand

• 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.

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