Question

Recall that the Fibonacci sequence {fn} is defined as follows: f1 = f2 = 1, and
Fn = fn−2 + fn−1, for n ≥ 3. to determine the convergence of the series
inf
Σ 1/Fn
n=1

so i know that the limit of fibonacci sequence goes to the golden ratio but how can i use that to solve for this? should i use some sort of comparison test?

Answer 1

shouldnt it converge?

Answer 2

yes, since the ratio test approaches 1/φ < 1

Answer 3

Yes, you can use the Comparison Test to determine the convergence of the series Σ 1/Fn.

Since the Fibonacci sequence is defined as Fn = fn−2 + fn−1 for n ≥ 3, we can see that Fn > fn−1, for n ≥ 3. Therefore, 1/Fn < 1/fn−1, for n ≥ 3.

Now, we know that the series Σ 1/fn converges if the ratio fn/fn−1 approaches a limit L < 1 as n approaches infinity. As you mentioned, the limit of the ratio of consecutive Fibonacci numbers approaches the golden ratio φ, which is greater than 1. Therefore, the series Σ 1/fn diverges.

Using the Comparison Test, we can conclude that since 1/Fn < 1/fn−1 and the series Σ 1/fn−1 diverges, the series Σ 1/Fn also diverges.