# Idea As $n$ approaches infinity, the [[limit]] is defined by the number $e$: $(1 + \frac{1}{n})^n$ $(1 + \frac{1}{10000})^{10000} = 2.71814593$ It is related to growth and compounding. I have $100 and my bank offers 100% interest per year. After a year, I have this amount: $100(1+1) = 100(2) = 200$ If instead the bank compounds the money 2 times a year (50% each time), after a year, I have this amount: $100(1+0.5)(1+0.5) = 100(1.5)^2 = 225$ If my bank compounds my money 4 times a year, after a year, I have this amount: $100(1+0.25)(1+0.25)(1+0.25)(1+0.25) = 100(1.25)^4 = 244.14$ This formula simplifies to the following: $100(1 + \frac{1}{n})^n$ It is known as [[continuous compounding]] in the banking world. As $n$ approaches infinity, we approach $e$: $(1 + \frac{1}{10000})^{10000} = 2.71814593$ That is, if we have $100 and get 100% interest on that per year, and we take the continuous compounding to the limit—31536000 time each year (i.e., once per second), we get this amount after a year: $100(1 + \frac{1}{31536000})^{31536000} = 100 * 2.71828178 = 271.828178$ # References - [[Strogatz 2019 infinite powers - how calculus reveals the secrets of the universe]]