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?
shouldnt it converge?
yes, since the ratio test approaches 1/φ < 1
To determine the convergence of the series Σ 1/Fn as n approaches infinity, we can indeed use a comparison test. In this case, the comparison test we can apply is the Direct Comparison Test.
The Direct Comparison Test states that if 0 ≤ a(n) ≤ b(n) for all n and the series Σ b(n) converges, then the series Σ a(n) also converges. Conversely, if 0 ≤ b(n) ≤ a(n) for all n and the series Σ b(n) diverges, then the series Σ a(n) also diverges.
In this case, let's compare the series Σ 1/Fn to another series that is known to converge. Since you mentioned that the Fibonacci sequence approaches the golden ratio, we can use this information to make a comparison.
The Fibonacci sequence approaches the golden ratio (φ) as n approaches infinity. The golden ratio is approximately 1.61803. Thus, the terms of the Fibonacci sequence (Fn) become larger as n increases.
Now, we can compare 1/Fn to 1/n^2. As n increases, the terms of the series 1/n^2 become smaller. It can be shown that the series Σ 1/n^2 converges (known as the Basel problem).
Since the terms of Σ 1/n^2 are larger than the terms of Σ 1/Fn, and Σ 1/n^2 converges, we can conclude that Σ 1/Fn also converges by the Direct Comparison Test.
Therefore, the series Σ 1/Fn converges.
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.