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Compute some properties of compound interest rates

Consider a bank account which starts with P dollars and pays an interest rate r per year, compounding m times per year.

  1. Prove that the balance in the bank account at the end of n years is

        \[ P \left( 1 + \frac{r}{m} \right)^{mn}. \]

For fixed values of r and n, the balance at the end of n years as m \to +\infty is given by

    \[ \lim_{m \to +\infty} P \left( 1 + \frac{r}{m} \right)^{mn} = Pe^{rn}. \]

We say that money grows at the annual rate r with continuous compounding if the amount of money after t years is denoted by f(t) is given by

    \[ f(t) = f(0)e^{rt}. \]

Give an approximate length of time for the money in a bank account to double if r = .06 and compounds:

  1. continuously;
  2. four times per year.

Incomplete. Sorry, I’ll try to get back to this soon(ish).

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