Convergence test

Calculus Level pending

Given that $p$ is fixed number, $f_n$ is the $n$ th Fibonacci number and $\displaystyle\sum_{n=1}^{\infty} \dfrac{f_n}{p^n}$ is convergent.

What are the possible values of $p$ ?

I. $p=1.5$

II. $p = 1.6$

III. $p = 1.7$

IV. $p = 1.8$

none of the options I, II, III and IV I and II only II, III, IV only I, II and III only III and IV only

1 solution

Will van Noordt
Dec 31, 2017

As $n\to \infty$ , we have $f_n \approx \phi^n$ , where $\phi$ is the golden ratio $\phi \approx 1.618...$

For $\sum \frac{f_n}{p^n}$ , we require $p > \phi$ . Hence, options 3 and 4 are valid.