Why can e, the number you get by doing 100% compound interest over infinite time periods, also be calculated by summing (1/x!) to infinity? What's the correlation/what does compound interest have to do with inverses of factorials?

Hi,

When you compound 100% annual interest over n periods of 1/n of a year, your money is multiplied by a\_n = (1 + 1/n)\^n. The number e, according to one definition, is the limit approached by this expression as n tends to infinity. You would like to know why this number e is also expressible as 1 + 1 + 1/2! + 1/3! + 1/4! + ...

Often the answer to this question is given via calculus and Taylor series, but I'm going to give a more elementary answer. The key is to rewrite a\_n by using the binomial formula. I'll write C(n,k) for the binomial coefficient n!/k!(n-k)!.

Then

a\_n = C(n,0) + C(n,1)/n + C(n,2)/n\^2 + ... + C(n,k)/n\^k + ... + C(n,n)/n\^n.

After some simplification and rewriting, this is

a\_n = 1 + 1 + (1/2!)(1 - 1/n) + (1/3!)(1 - 1/n)(1 - 2/n) + ... + (1/k!)(1 - 1/n)(1 - 2/n)...(1 - (k-1)/n) + ... + (1/n!)(1 - 1/n)(1 - 2/n)...(1 - (n-1)/n)).

As n increases, two things happen to this expression. First, the number of terms increases indefinitely. And second, if you examine just the (k+1)st term for a fixed value of k, the factors 1 - 1/n, 1 - 2/n, ..., 1 - (k-1)/n, all tend to 1, so the term itself approaches 1/k!.

It is plausible therefore to assert that a\_n approaches the sum 1 + 1 + 1/2! + 1/3! + 1/4! + ... as n tends to infinity.

To turn this argument into a correct proof would take some additional work. We would need to show why the fact that there are an ever-increasing number of terms in a\_n doesn't change our conclusion. But in fact, it is possible to do this by showing that these terms are "small" in a precise sense I won't go into.

Edit. For those who do want a full proof, note first that in fact a\_n is bounded above by 1 + 1 + 1/2! + 1/3! + ... Moreover, as all the terms in a\_n are positive, the argument we've just given shows that lim inf a\_n >= 1 + 1 + 1/2! + ... + 1/k! for any fixed k we choose. That is enough to conclude that lim a\_n = 1 + 1 + 1/2! + 1/3! + ...
The number e itself isn't particularly important or interesting, it's just the exponential function evaluated at 1. The function is what's important.

If you want to understand where the factorial representation comes from look into taylor series.

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