MHB Finding Limit on 58(b): UGA Math 115 Homework #2

[ineedhelpnow]

is there any other way to find the limit on 58(b). its on page 3 of the link.

http://www.math.uga.edu/~clayton/teaching/m115f09/homework/hw2solutions.pdf

 

[Euge]

The only other way I can think of is to verify by induction that $a_n = F_n/F_{n+1}$ ($F_n$ denotes the $n$th term of the Fibonacci sequence), then use the identity

$F_n = \frac{1}{\sqrt{5}}\left\{\left(\frac{1+\sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n\right\}$

to compute $\lim_{n\to \infty} a_n$ directly.