1
vote
1answer
63 views

Limit of a function not using Stirling's Approximation

I want to compute the following limit: $$\lim_{n\to\infty} \frac{\left(\frac{e}{F_{n+1}}\right)^{F_{n+1}} F_{n+1}!}{\left(\frac{e}{F_n}\right)^{F_n} F_n!},$$ where $F_n$ is the $n$th Fibonacci ...
0
votes
3answers
142 views

A sum involving Fibonacci numbers, $\sum_{k=1}^\infty F_k/k!$

Let $F_k$ be Fibonacci numbers. I am looking for a closed form of the sum $\sum_{k=1}^\infty F_k/k!$. I tried to use Wolfram Alpha, but it is not doing the sum Fibonacci[k]/k! , k=1 to infinity. ...
2
votes
1answer
221 views

Proof the following proposition: for all $n \geq 0, \mathrm{fib}(n) \leq n!$

I am a comp science undergrad and just started to learn proof. And I have been thinking about this question for a few days. How should I present my answer? Do I have to use the Binet's formula? Or can ...
7
votes
1answer
713 views

How to prove that $\mathrm{Fibonacci}(n) \leq n!$, for $n\geq 0$

I am trying to prove it by induction, but I'm stuck $$\mathrm{fib}(0) = 0 < 0! = 1;$$ $$\mathrm{fib}(1) = 1 = 1! = 1;$$ Base case n = 2, $$\mathrm{fib}(2) = 1 < 2! = 2;$$ Inductive case ...