The number of years \(n\) required for an investment at interest rate \(r\) to double in value must satisfy \((1+r)^{n}=2\). Using \(\ln 2=.69\) and the approximation \(\ln (1+r) \approx r\) valid for small \(r\), show that \(n \approx 69 / i\), where \(i\) is the interest rate percentage (that is, \(i=100 r\) ). Using the better approximation \(\ln (1+r) \approx r-\frac{1}{2} r^{2}\), show that for \(r \approx .08\) there holds \(n \approx 72 / i\).