J. Mike Rollins (Sparky) [rollins@wfu.edu]
  CISSP, GIAC GPEN
Hyperbola New
My Cats New
Kitty New
Mike is on
  LinkedIn
  FaceBook
BackYardGreen.Net
HappyPiDay.com
Green Cycle Design Group
CamoTruck.Net
  Resume  
  My Stuff  
  Art  
  My Truck  
  People  
Electronics
Jacob's Ladder
Scripts
Math
   Random Numbers
   Maple code
   Loan
   L'Hopital's rule
   Hyperbola New
   Fraction e^x
   Mandelbrot
Notes
My House
My Cars
My Cats New
My Jokes
Pi Poetry
pumpkin
Toro Mower
Development
Speed of a Piston
Not a Pipe
Linux
















Fraction e^x

Converting a Power Series to a Continued Fraction
with e^x as an Example

Iteration Process
e^x example

Back in 1989 or 1990, I started playing around with continued fractions. I was bored and decided to try to convert e^x from a power series to a continued fraction. I came to a very neat solution.

I then tried to formalize a way to convert any power series to a continued fraction. After having about 15 sheets of paper with scribbles all over the place, I decided I had not been that bored.

About five years later, I discovered Maple. I put in the general power series and wrote a couple of little Maple procedure. In less than a minute, I saw what had taken days to do by hand. I also found that many determinants lived inside this continued fraction. The following is a summary of what I found.

I have looked around the net for this fraction and have not seen it yet, so I thought I would put this web page out there. If you have any further information about this fraction, please send me an email, rollins@wfu.edu.




 
Iteration 1

Begin with a basic power series.

Factor out x from the second term to infinity.

Convert the parenthesized expression to a fraction.

Expand the denominator by performing long division.

Iteration 2

Next, factor out x from the series as we did before.

Convert the series to a fraction.

Expand the denominator by long division again.

Iteration 3

This is starting to get a little ugly. So, let's not do iteration 3. So, I will just jump to the interesting part.

The Pattern click here for a more general description




  A very elegant fraction can be derived for e to the x power.




Assign values for a0,a1,a2 ...




Next, we can evaluate the messy fractions of determinants.

I used Maple to evaluate these determinants. It gets really messy.



Now we can create our continued fraction with these values.




In this final step, I move the negative sign around so that the fraction has an alternation between + and -. I also have added 1 to both sides so that the value of 2 is also alternating.