Continued fractions and functions

[tpm]

for any function $$f(x)$$ how can you convert it to continued fractions in the form:

$$\frac{a_{0}+a_{1}x+a_{2}x^{2}+...}{b_{0}+b_{1}x+b_{2}x^{2}+...}$$

where we must determine the a(n) and b(n) if $$f(x)=x^{1/2}$$ i know how to do it.

 

[NateTG]

That's not actually what people usually call a continued fraction. For more on continued fractions you could refer to:
http:/mathworld.wolfram.com/ContinuedFraction.html

Functions can't always be expressed in that form.
http://mathworld.wolfram.com/WeierstrassFunction.html

Are you familiar with the Taylor-McLaurin series?

 

[tacman]

To convert a function f(x) to continued fractions, we can use the following steps:

1. Expand the function f(x) as a power series, which is a series of terms with increasing powers of x. This can be done using Taylor series or any other method of expansion.

2. Write the power series in the form of a fraction, with the coefficients of the terms as the numerator and the powers of x as the denominator.

3. Group the terms with the same powers of x together and simplify the resulting fraction.

4. The resulting fraction will be in the form of a continued fraction, with the coefficients of the terms in the numerator as the a(n) values and the coefficients of the powers of x in the denominator as the b(n) values.

For example, let's consider the function f(x) = x^(1/2). Using the above steps, we can expand the function as:

f(x) = 1 + (1/2)x - (1/8)x^2 + (1/16)x^3 - (5/128)x^4 + ...

Writing this as a fraction, we get:

f(x) = (1 + (1/2)x - (1/8)x^2 + (1/16)x^3 - (5/128)x^4 + ...)/(1)

Grouping the terms with the same powers of x together, we get:

f(x) = (1 + (1/2)x)/(1 - (1/8)x - (5/128)x^2 - ...)

This can be further simplified and written in the form of a continued fraction as:

f(x) = (1 + (1/2)x)/(1 - (1/8)x/(1 - (5/128)x/(1 - ...)))

In this form, the a(n) values are 1, 1/2, and the b(n) values are 1, 1/8, 5/128, and so on.

In conclusion, to convert a function f(x) to continued fractions, we can use the steps of expanding the function as a power series, writing it as a fraction, grouping the terms with the same powers of x, and simplifying the resulting fraction. This will give us the continued fraction in the form of (a(n))/(b(n)).