Find the value of this infinite decimal:That is: .Code:`0.01`

0.001

0.0002

0.00003

0.000005

0.0000008

0.00000013

0.000000021

+ ...

-----------------

S = 0.011235955 ...

. . where is the Fibonacci number.

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- June 23rd 2007, 03:16 PMSorobanFibonacci decimal
Find the value of this infinite decimal:

Code:`0.01`

0.001

0.0002

0.00003

0.000005

0.0000008

0.00000013

0.000000021

+ ...

-----------------

S = 0.011235955 ...

. . where is the Fibonacci number.

- June 23rd 2007, 03:46 PMtopsquark
- June 23rd 2007, 04:30 PMgalactus
Hey Soroban. I think I may have this one. Let's see. I encountered this proof as a homework assignment back in college. In a Seminars in Mathematics course.

[tex]\frac{1}{10}\sum_{n=1}^{\infty}\frac{F_{n}}{10^{n} }[\math]

Let

Now use and we have:

So, we have:

Don't forget to multiply by our 1/10 and get:

- June 23rd 2007, 07:02 PMThePerfectHacker
Welcome back, you were away for a long time!

- June 24th 2007, 03:33 AMgalactus
Thanks PH. Yeah, it has been a while. I see you're up to 6200. Very prolific:)

How's the academic world treating you?. - June 24th 2007, 07:04 AMThePerfectHacker
- June 24th 2007, 05:13 PMGoistein
Quote

http://www.mathhelpforum.com/math-he...94c654a9-1.gif

/Quote

I'm pretty sure that F(4)/10 should be F(4)/100, etc.;) - June 25th 2007, 11:09 AMgalactus
You're so right, G. Thanks for the catch. I fix.

- June 25th 2007, 01:42 PMtopsquark
- June 25th 2007, 02:06 PMgalactus
Cool, Topsquark. That's a good one.

One could also just use the closed-form for the Fibonacci numbers with x=1/10.

Multiply by our 1/10 and we get 1/89.