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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.
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Now we can create our continued fraction with these values.
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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.
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