* Original value: 1 (this is our starting amount, and can be written 1/0! = 1)
* First-level interest: 1 (this is our basic 100% return, and can be written 1/1! = 1)
* Second-level interest: 1/2! (this is the interest on our 100% return, which is the integral of “x”: 1/2 * x^2 where x = 1)
* Third-level interest: 1/3! (interest on our second-level interest (1/2 x^2) is 1/3 * 1/2 x^3 = 1/6 x^3 = 1/6)

and so on. Basically, e^x can be seen as

1 + x + x^2/2! + x^3/3! + x^4/4! + ….

and each term is computing the interest on the term before it by taking its integral. We plug in x = 1 to get e^1 = e. Hope this helps! (I should write a follow up on this, it’s a good point). Also, see the article on sine to see another example of this “interest on interest” pattern: http://betterexplained.com/articles/intuitive-understanding-of-sine-waves/

Great website!! Just a quick question….with regard to your interesting explanation of e’s Taylor series, could you maybe explain to me the math as to why the interest of the interest is 1/2! and the interest of the interest of the interest is 1/3! and so on and so forth? I see the principal value and the direct interest in the series, but after that, I’m a little lost as to why those next terms are the mathematical expressions of the subsequent interests…..

Thanks so much!!

Sincerely,
Stephen