Chapter 8.4, Problem 73E

a.

To determine

To show: The value $\phi \approx 1.618$.

b.

To determine

To show: $\underset{n\to \infty }{\lim }\frac{f_{n-1}}{f_{n+1}}=1-\frac{1}{\phi }\approx 0.382$.

c.

To determine

To show: ${\displaystyle \sum _{k=1}^{\infty }\frac{1}{k}}\ge 1+{\displaystyle \sum _{k=1}^{\infty }\frac{f_{k-1}}{f_{k+1}}}$ using Fibonacci sequence.

d.

To determine

To conclude: The harmonic series diverges using part (b).