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     x * ( 1 *1 *1 ... *1) +  (1) * x *( 1*1 *1 ,,,,*1) + ...... ( 1* 1 *1 * 1 .... * 1) * x    with n positions of x meaning  the result is  x*n.
  • degree two term     x *  x * ( 1 *1 *1 ... *1) +   x * (1) * x *( 1*1 *1 ,,,,*1) + ...... x * ( 1* 1 *1 * 1 .... * 1) * x    with n-1  positions of x meaning  the result is  x*2 *n!. 

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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2 Comments

Prof. Von Nostrand MonsterID Icon Prof. Von Nostrand · April 7, 2022 at 9:54 am

Nice.  Can’t ya use LaTeX to write yer equations on yer blog like I do:

https://phxmarker.blogspot.com/2021/05/just-for-latex-practice-defintion-of.html
Right click on the equation to see MathJax info etc

& LaTeX reference:

https://en.wikibooks.org/wiki/LaTeX/Mathematics

Fudgy McFarlen MonsterID Icon Fudgy McFarlen · April 11, 2022 at 5:51 pm

There is LaTex in this post. But I can not in the title so far as I know.

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