Number e and compound interest
Suppose you invest some amount [math]\displaystyle{ P }[/math] for a period of time (that can be a week, month, year etc.) at effective rate of compound interest [math]\displaystyle{ i. }[/math] At the end of one such period you have [math]\displaystyle{ P +iP = P(1+i) }[/math] accumulated in your account. If you keep your investment for two such periods, the value in your account will be [math]\displaystyle{ P(1+i) + i[P(1+i)] = P(1+i)^2. }[/math] In general, if the investment is for [math]\displaystyle{ n }[/math] periods, at the end your account will have accumulated [math]\displaystyle{ P(1+i)^n. }[/math]
Suppose, for simplicity, that one period of investment is one year. Suppose further that you invest [math]\displaystyle{ P = 1 }[/math] for the period of one year, and the rate of interest is 100% or, equivalently, [math]\displaystyle{ i = 1 }[/math] (this means that the amount you invested will double at the end of the year, which, of course, rarely occurs in practice). Alternatively, you