Taylor polynomial of f(x) = 1/(1-x) and the estimate of its remainder

In summary, the conversation discusses finding the Taylor polynomial for f(x) = 1/(1-x), with n=5 and centered around 0. An estimate of the remainder is also given using the Lagrange form, with a maximum upper bound of 2. There is some confusion about the choice of interval for the estimate, but ultimately it is determined that showing the calculation process is more important.
  • #1
Ryker
1,086
2

Homework Statement


Find the Taylor polynomial for f(x) = 1/(1-x), n = 5, centered around 0. Give an estimate of its remainder.

The Attempt at a Solution


I found the polynomial to be 1 + x + x2 + x3 + x4 + x5, and then tried to take the Lagrange form of the remainder, say, for x in [-1/2, 1/2].

Then I get
[tex]R_{5}(x) = \frac{f^{6}(a)}{6!}x^{6}[/tex]

But
[tex]f^{6}(a) = \frac{720}{(1-a)^{7}},[/tex]
so for the largest values of a and x, that is 1/2, you get that
[tex]|R_{5}(x)| \leq \frac{(\frac{1}{2})^{6}}{(\frac{1}{2})^{7}} = 2,[/tex]
which means the best estimate we can give is that the remainder is less than 2. But that doesn't make much sense to me, because the 5-th degree polynomial is fairly accurate on [-1/2, 1/2] and really close to f(x), so how I am getting such results? Have I made a mistake somewhere?
 
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  • #2
That looks fine to me. The remainder 'estimate' isn't really an estimate of the remainder. It's a maximum upper bound for the remainder. In this case it's a pretty pessimistic estimate, probably because you're nearing the singularity at x=1.
 
  • #3
Hey, thanks for the reply. Yeah, I figured it's only the maximum upper bound, so should I have taken a smaller interval? I mean, we aren't really given any specific range, so I guess we can choose our own, it's just that I didn't want to choose one, smaller than [-1/2, 1/2]. Or am I right in thinking none of this really matters, as long as you show how you got to that estimate?
 
  • #4
Ryker said:
Hey, thanks for the reply. Yeah, I figured it's only the maximum upper bound, so should I have taken a smaller interval? I mean, we aren't really given any specific range, so I guess we can choose our own, it's just that I didn't want to choose one, smaller than [-1/2, 1/2]. Or am I right in thinking none of this really matters, as long as you show how you got to that estimate?

I don't see anything wrong with what you've got. Maybe they just want the expression for R_5, not a specific number.
 
  • #5
I put down both just in case anyway, so I guess I should be covered then :smile: Thanks again.
 

Related to Taylor polynomial of f(x) = 1/(1-x) and the estimate of its remainder

1. What is a Taylor polynomial and how is it used to approximate a function?

A Taylor polynomial is a mathematical expression that represents a function as a finite sum of terms, each term being a polynomial function of the variable. It is used to approximate a function by using a series of polynomial equations to better understand its behavior and make predictions about its values at different points.

2. How is the Taylor polynomial of f(x) = 1/(1-x) calculated?

The Taylor polynomial of f(x) = 1/(1-x) is calculated using the Taylor series expansion, which is a way of representing a function as an infinite sum of terms. In this case, the Taylor series is centered at x = 0, and the coefficients are determined by taking derivatives of the function at x = 0.

3. What is the remainder term in the Taylor polynomial of f(x) = 1/(1-x)?

The remainder term in the Taylor polynomial of f(x) = 1/(1-x) represents the difference between the actual value of the function and the approximate value given by the Taylor polynomial. It is usually written in terms of the maximum value of the (n+1)th derivative of the function within the interval of interest, where n is the degree of the Taylor polynomial.

4. How is the estimate of the remainder calculated in the Taylor polynomial of f(x) = 1/(1-x)?

The estimate of the remainder in the Taylor polynomial of f(x) = 1/(1-x) is calculated using the Lagrange remainder formula, which is a way of approximating the remainder term. The formula involves taking the (n+1)th derivative of the function and evaluating it at a point within the interval of interest. The value obtained from this calculation is then multiplied by (x-a)^(n+1), where a is the center of the Taylor polynomial.

5. How accurate is the Taylor polynomial of f(x) = 1/(1-x) in approximating the function?

The accuracy of the Taylor polynomial of f(x) = 1/(1-x) depends on the degree of the polynomial and the value of x at which it is evaluated. Generally, the more terms included in the polynomial, the more accurate the approximation will be. However, as x moves farther away from the center of the polynomial, the accuracy may decrease. The estimate of the remainder can also give an indication of the accuracy of the polynomial at a specific point.

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