Continued Fractions Approximation

$\frac{{x}^{2}+3x+2}{{x}^{2}-x+1}$

$\frac{{x}^{2}+3x+2}{{x}^{2}-x+1}$

uri2e4g
2022-07-01
Answered

Continued Fractions Approximation

$\frac{{x}^{2}+3x+2}{{x}^{2}-x+1}$

$\frac{{x}^{2}+3x+2}{{x}^{2}-x+1}$

You can still ask an expert for help

Alisa Jacobs

Answered 2022-07-02
Author has **13** answers

$\begin{array}{rl}\frac{{x}^{2}+3x+2}{{x}^{2}-x+1}& =1+\frac{{\textstyle 4x+1}}{{\textstyle {x}^{2}-x+1}}\\ & =1+\frac{{\textstyle 1}}{{\textstyle \frac{1}{4}x-\frac{5}{16}+\frac{{\textstyle \frac{21}{16}}}{{\textstyle 4x+1}}}}\\ & =1+\frac{{\textstyle 1}}{{\textstyle \frac{1}{4}x-\frac{5}{16}+\frac{{\textstyle 1}}{{\textstyle \frac{64}{21}x+\frac{16}{21}}}}}\end{array}$

At each stage, we are doing a polynomial division instead of an integer division, but otherwise, the process is the same as with continued fractions with integers.

We can get the Bezout polynomials by truncating the continued fraction:

$1+\frac{{\textstyle 1}}{{\textstyle \frac{1}{4}x-\frac{5}{16}}}=\frac{4x+11}{4x-5}$

That is, we can write the polynomial GCD (a constant since they are relatively prime) as

$(4x+11)({x}^{2}-x+1)-(4x-5)({x}^{2}+3x+2)=21$

At each stage, we are doing a polynomial division instead of an integer division, but otherwise, the process is the same as with continued fractions with integers.

We can get the Bezout polynomials by truncating the continued fraction:

$1+\frac{{\textstyle 1}}{{\textstyle \frac{1}{4}x-\frac{5}{16}}}=\frac{4x+11}{4x-5}$

That is, we can write the polynomial GCD (a constant since they are relatively prime) as

$(4x+11)({x}^{2}-x+1)-(4x-5)({x}^{2}+3x+2)=21$

asked 2022-07-09

How can I get an approximation formula for the sum $J(n)={2}^{-n}\sum _{k=1}^{n}\frac{1}{k}{\textstyle (}\genfrac{}{}{0ex}{}{n}{k}{\textstyle )}$?

asked 2022-05-03

A function of $2$ variable is given by,

$f(x,y)={e}^{2x-3y}$

How to find tangent approximation to $f(0.244,1.273)$ near $(0,0)?$?

$f(x,y)={e}^{2x-3y}$

How to find tangent approximation to $f(0.244,1.273)$ near $(0,0)?$?

asked 2022-09-06

Let p represent a false statement, let q represent a false statement, and let r represent a false statement. Find the truth value of the given statement.

$r\to \sim p$

Is the statement true or false?

$r\to \sim p$

Is the statement true or false?

asked 2022-07-04

Let $f(x)=arctan(x)$. Use the derivative approximation:

${f}^{\prime}(x)=\frac{8f(x+h)-8f(x-h)-f(x+2h)+f(x-2h)}{12h}$ to approximate ${f}^{\prime}(\frac{1}{4}\pi )$ using ${h}^{-}1$ = $2,4,8$ . Try to take $h$ small enough that the rounding error effect begins to dominate the mathematical error. For what value of h does this begin to occur?

${f}^{\prime}(x)=\frac{8f(x+h)-8f(x-h)-f(x+2h)+f(x-2h)}{12h}$ to approximate ${f}^{\prime}(\frac{1}{4}\pi )$ using ${h}^{-}1$ = $2,4,8$ . Try to take $h$ small enough that the rounding error effect begins to dominate the mathematical error. For what value of h does this begin to occur?

asked 2022-09-03

Let p represent a false statement, let q represent a true statement, and let r represent a true statement. Find the truth value of the given statement.

$\sim r\to (p\wedge \sim q)$

Is the statement true or false?

$\sim r\to (p\wedge \sim q)$

Is the statement true or false?

asked 2022-07-25

The home range, in hectares, of a carnivorous mammal weighing w grams can be approximated by $H(w)=0.11{w}^{1.36}$

a)Find the average rate at which a carnivorous mammal's home range increases as the animal's weight grows from 200g to 450g.

b) Find $\frac{H(500)-H(400)}{500-400}$, and interpret this result.

a)Find the average rate at which a carnivorous mammal's home range increases as the animal's weight grows from 200g to 450g.

b) Find $\frac{H(500)-H(400)}{500-400}$, and interpret this result.

asked 2022-06-24

Of the following, which is the best approximation of

$\sqrt{1.5}(266{)}^{3/2}$

$\sqrt{1.5}(266{)}^{3/2}$