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I am trying to prove the convergence/divergence of the series´ $$\sum_{i=1}^\infty \frac{1}{F_n} = 1+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+...$$ ${F_n}$ being the Fibonacci sequence.
The Fibonacci sequence is defined without recursion by: $${F_n}=\frac{\phi^n-(-\phi)^{-n}}{\sqrt{5}} \quad\land\quad\phi=\frac{1+\sqrt{5}}{2} $$ I have tried to prove its convergence with the Root Test and the Ratio Test because of the $n$ exponent but can't manage to do it because of the difference in the fraction.

Can anyone help me? Thank you

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    $\begingroup$ Both the root and ratio tests work. $\endgroup$ – Lord Shark the Unknown May 23 '17 at 19:42
  • $\begingroup$ I'm sure they do, but I'm having problems in the limit calculation because of the difference. If someone could hint me in the right direction it would be great! Thank you. $\endgroup$ – Luis Dias May 23 '17 at 19:45
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    $\begingroup$ Use the usual trick of pulling out the dominant term: $F_n=\phi^n a_n$ where $a_n\to1/\sqrt5$. $\endgroup$ – Lord Shark the Unknown May 23 '17 at 19:46
  • $\begingroup$ Thank you so much! The exercises I was doing didn't require much limit calculation techniques and when I tried to prove something on my own I didn't even remember the old tricks! $\endgroup$ – Luis Dias May 23 '17 at 19:54
  • $\begingroup$ Maybe you find the answer of this post useful. $\endgroup$ – Amin235 May 25 '17 at 20:13
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Since $$F_{n}=\frac{\phi^{n}-\left(-\phi\right)^{-n}}{\sqrt{5}}$$ we have $$F_{n}\sim\frac{\phi^{n}}{\sqrt{5}}$$ as $n\rightarrow\infty$ so $$\sum_{n\ge1}\frac{1}{F_{n}}\sim\sqrt{5}\sum_{n\ge1}\frac{1}{\phi^{n}}=\frac{\sqrt{5}}{\phi-1}.$$

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You may prove by induction that for any $n\geq 5$ we have $F_{n+5}\geq 11 F_n$. That is enough to deduce convergence by comparison with a geometric series and further get that:

$$ S = \sum_{n\geq 1}\frac{1}{F_n} =\frac{17}{6}+\sum_{n\geq 5}\frac{1}{F_n} = \frac{17}{6}+\sum_{n=5}^{9}\frac{1}{F_n}+\sum_{n\geq 10}\frac{1}{F_n}\leq \frac{17}{6}+\frac{88913}{185640}+\frac{1}{11}\sum_{n\geq 5}\frac{1}{F_n}$$ such that: $$ \frac{10}{11}\sum_{n\geq 5}\frac{1}{F_n}\leq \frac{88913}{185640},\qquad S\leq \frac{17}{6}+\frac{11}{10}\cdot \frac{88913}{185640}=\frac{2079281}{618800}. $$

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For $n \ge 3$, we have

$$\frac{1}{F_n} \le \frac{F_{n-1}}{F_nF_{n-2}} = \frac{F_{n} - F_{n-2}}{F_nF_{n-2}} = \frac{1}{F_{n-2}} - \frac{1}{F_n} = \left(\frac{1}{F_{n-2}} + \frac{1}{F_{n-1}}\right) - \left(\frac{1}{F_{n-1}} + \frac{1}{F_n}\right)\\ = \frac{F_{n}}{F_{n-2}F_{n-1}} - \frac{F_{n+1}}{F_{n-1}F_n} $$

The partial sums is monotonic increasing and bounded from above. $$\sum_{n=1}^N \frac{1}{F_n} = 2 + \sum_{n=3}^N \frac{1}{F_n} \le 2 + \frac{F_3}{F_1F_2} - \frac{F_{N+1}}{F_{N-1}F_N} \le 2 + \frac{2}{1\cdot 1} = 4$$ As a result, the series converges.

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$$\frac{F_{n+1}}{F_n}=\frac{\phi^{n+1}-(-\phi)^{-n-1}}{\phi^n-(-\phi)^{-n}}\ge\phi\frac{1-\phi^{-2n-2}}{1+\phi^{-2n}}.$$

The expression on the right is an increasing function of $n$ that exceeds $1$ as of $n=2$ (and quickly tends to $\phi$).

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Let $F_n$ be the fibonacci sequence

We know $F_{2n+2}=F_{2n+1}+F_{2n}\geq 2F_{2n}$

Similarly $F_{2n+1}=F_{2n}+F_{2n-1}\geq 2F_{2n-1}$

So $1/F_2+1/F_4+1/F_6\dots<1/F_2+1/2F_2+1/4F_2\dots=1/F_2(1/2+1/4+1/8\dots)$

$1/F_1+1/F_3+1/F_5\dots<1/F_1+1/2F_1+1/4F_1\dots=1/F_1(1/2+1/4+1/8\dots)$

Now I think it's clearly evident that why the sum of reciprocals of the Fibonacci sequence is convergent, only the definition of the Fibonacci sequence is enough!

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Suppose $2 \ge \frac{F_{n+1}}{F_n} \ge \frac32 $ for $n$ and $n+1$.

Then

$\frac{F_{n+2}}{F_{n+1}} =\frac{F_{n+1}}{F_{n+1}}+\frac{F_{n}}{F_{n+1}} =1+\frac{F_{n}}{F_{n+1}} \ge 1 + \frac12 =\frac32 $

and

$\frac{F_{n+2}}{F_{n+1}} =\frac{F_{n+1}}{F_{n+1}}+\frac{F_{n}}{F_{n+1}} =1+\frac{F_{n}}{F_{n+1}} \lt 1 + 1 =2 $.

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I think I have an interesting proof for the summability of the reciprocals of the FN. It is based on the Kummer's test and the fact that $F_{n-1}\geq 2 F_{n}$, $n\geq 2$. First note that $(1/F_n)$ is summable, if and only if, there exists a sequence $(u_n)$ of positive numbers such that $$u_n F_{n+1}-u_{n+1}F_{n}\geq F_{n},$$ for all $n$ sufficiently large. (This is the Kummer's test applied to the sequence $(1/F_n)$). Now, from the definition of FN: $F_{n+1}-F_{n}=F_{n-1}$, for all $n\geq 2$.

That is, taking $u_n = 2$, for all $n\geq 1$, we have that $2 F_{n+1}- 2 F_{n}= 2 F_{n-1}\geq F_{n}$, for all $n\geq 2$. As so, by the Kummer's test, the sequence $(1/F_n)$ is summable.

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