Bounding the Error in Taylor Series Approximations for ln(1+x)

In summary, the conversation discusses finding a bound for the error in using a third-order Taylor polynomial to approximate ln(1+x) on the interval [-1/2,1/2]. The Taylor series of ln(1+x) is provided and the formula for the remainder after n = 3 is mentioned. The speaker is seeking help in determining the maximum value of the remainder for x and c within the given interval.
  • #1
francisg3
32
0
Had a recent homework questions:
Find a bound for the error |f(x)-P3(x)| in using P3(x) to approximated f(x) on the interval [-1/2,1/2]
where f(x)=ln(1+x) abd P3(x) refers to the third-order Taylor polynomial.

I found the Taylor series of f(x) seen below:

x- x^2/2!+(2x^3)/3!

I know the Taylor series expression has a remainder which in this case would be the 4th order polynomial and beyond but I am completely lost beyond this. Any help would be greatly appreciated!
 
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  • #2
Your formula for the remainder after n = 3 is

[tex]\frac{f^{(4)}(c)}{4!}(x-a)^4[/tex]

Your a = 0. How large in absolute value can this be for x and c in your given interval?
 

FAQ: Bounding the Error in Taylor Series Approximations for ln(1+x)

1. What is the Taylor Series Error?

The Taylor Series Error is the difference between the actual value of a function and the value calculated using the Taylor series approximation. It represents the accuracy of the approximation and can be used to determine the number of terms needed for a desired level of precision.

2. How is the Taylor Series Error calculated?

The Taylor Series Error is calculated using the remainder term of the Taylor series, which is given by the formula: R_n(x) = f^(n+1)(c) * (x-a)^(n+1) / (n+1)! where n is the number of terms used in the Taylor series, x is the value at which the approximation is being made, a is the center of the series, and c is a value between x and a.

3. Why is the Taylor Series Error important?

The Taylor Series Error is important because it allows us to estimate the accuracy of a Taylor series approximation. This is useful in many areas of mathematics and science, as it allows us to determine the number of terms needed for a desired level of precision and to evaluate the validity of mathematical models and equations.

4. How can the Taylor Series Error be reduced?

The Taylor Series Error can be reduced by increasing the number of terms used in the Taylor series, as this will result in a more accurate approximation. Additionally, choosing a center point closer to the value at which the approximation is being made can also help reduce the error.

5. Are there any limitations to using the Taylor Series Error?

Yes, there are some limitations to using the Taylor Series Error. This method of approximation assumes that the function is differentiable an infinite number of times, which may not always be the case. Additionally, the series may not converge for certain values of x, making the approximation invalid. It is important to carefully consider these limitations when using the Taylor Series Error in any calculation.

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