# Fibonacci Power Series

• Mar 15th 2011, 01:30 PM
worc3247
Fibonacci Power Series
I am having trouble with this question:
Define $\displaystyle F(x)=$\displaystyle\sum\limits_{n=0}^\infty F_nx^n$$where \displaystyle F_n is the nth Fibonacci number. Determine the radius of convergence of F(x). Using \displaystyle F_{n+2} = F_{n+1} + F_n find F(x) in closed form. I think I have managed to do the radius of convergence but I don't know if I am correct. Is the answer the solution to \displaystyle x^2-x-1 i.e. \displaystyle \frac{2}{1+\sqrt{5}}? As for the second part I have completely no idea and would appreciate some help starting the question (I should note I have tried writing F(x) out in terms of the formula but have not gotten anywhere with this) • Mar 15th 2011, 02:11 PM chisigma Quote: Originally Posted by worc3247 I am having trouble with this question: Define \displaystyle F(x)=\displaystyle\sum\limits_{n=0}^\infty F_nx^n$$ where $\displaystyle F_n$ is the nth Fibonacci number. Determine the radius of convergence of F(x). Using $\displaystyle F_{n+2} = F_{n+1} + F_n$ find F(x) in closed form.

I think I have managed to do the radius of convergence but I don't know if I am correct. Is the answer the solution to $\displaystyle x^2-x-1$ i.e. $\displaystyle \frac{2}{1+\sqrt{5}}$?

As for the second part I have completely no idea and would appreciate some help starting the question (I should note I have tried writing F(x) out in terms of the formula but have not gotten anywhere with this)

The explicit form for the Fibonacci's number is...

$\displaystyle \displaystyle F_{n}= \frac{\varphi^{n} - (-1)^{n}\ \varphi^{-n}}{\sqrt{5}}$ (1)

... where...

$\displaystyle \displaystyle \varphi= \frac{1+\sqrt{5}}{2}$ (2)

... so that is...

$\displaystyle \displaystyle F(x)= \sum_{n=0}^{\infty} F_{n}\ x^{n} = \frac{1}{\sqrt{5}}\ \{\sum_{n=0}^{\infty} (\varphi\ x)^{n} - \sum_{n=0}^{\infty} (-\frac{x}{\varphi})^{n}\}$ (3)

Now (3) is the sum of two series... the first converges for $\displaystyle |x|<\varphi^{-1}$ , the second for $\displaystyle |x|<\varphi$ and is $\displaystyle \varphi^{-1}< \varphi$ so that...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Mar 16th 2011, 05:21 AM
worc3247
Do you then just take geometric series and get:
$\displaystyle F(x)=\frac{1}{\sqrt{5}}(\frac{1}{1-\varphi x} - \frac{\varphi}{\varphi + x})$?
Cheers
• Mar 16th 2011, 06:50 AM
chisigma
Quote:

Originally Posted by worc3247
Do you then just take geometric series and get:
$\displaystyle F(x)=\frac{1}{\sqrt{5}}(\frac{1}{1-\varphi x} - \frac{\varphi}{\varphi + x})$?
Cheers

Yes!... it does!...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$