Lagrange Remainder for Taylor Expansion of ln(4/5) ≤ 1/1000?

In summary, the conversation is about trying to limit Lagrange's remainder on the Taylor expansion of ln(4/5) to be less than or equal to 1/1000. The person has tried using both ln(1+x) and ln(x) with different values for x and x0, but keeps getting a result that leads to n ≥ 3. However, upon expanding up to the third power, the result has an error greater than 1/1000. The person is seeking advice and wondering if there is something wrong with their formulation of the problem. They also mention that they have not received any replies and ask for help with their working so far.
  • #1
peripatein
880
0
Hi,

Homework Statement


I am trying to limit Lagrange's remainder on taylor expansion of ln(4/5) to be ≤ 1/1000.


Homework Equations





The Attempt at a Solution


I have tried using both ln(1+x), where x=-1/5 and x0(the center)=0, and ln(x), where x=4/5 and x0=1.
Every time I keep getting that (n+1)4n+1≥1000, leading to n ≥ 3.
But then, upon expansion up to the third power, I keep getting a result whose error is greater than the desired 1/1000. It appears the result should have been n≥4, but why so when algebra seems to prove it not to be the case?
I'd appreciate some advice. Obviously I am missing something.
 
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  • #2
I am rather surprised no one has replied. Is there anything amiss with my formulation of the problem?
 
  • #3
peripatein said:
Every time I keep getting that (n+1)4n+1≥1000, leading to n ≥ 3.
Please post your working to that point.
 

Related to Lagrange Remainder for Taylor Expansion of ln(4/5) ≤ 1/1000?

1. What is the Lagrange remainder question?

The Lagrange remainder question is a mathematical problem in calculus that involves estimating the error or remainder when using a polynomial to approximate a function. It is named after Joseph-Louis Lagrange, a famous mathematician who developed the theorem that is used to solve this problem.

2. Why is the Lagrange remainder question important?

The Lagrange remainder question is important because it allows us to determine the accuracy of polynomial approximations. This can be useful in various applications, such as engineering, physics, and economics, where approximations are commonly used to simplify complex problems.

3. How is the Lagrange remainder calculated?

The Lagrange remainder is calculated using the Lagrange error bound formula, which is given by Rn(x) = |f(n+1)(c)| * (x-a)^(n+1) / (n+1)!, where f(n+1)(c) is the (n+1)th derivative of the function at some point c between x and a. This formula gives an upper bound for the error or remainder when using a polynomial of degree n to approximate a function at a specific point x.

4. What are the assumptions for using the Lagrange remainder?

The Lagrange remainder formula assumes that the function is continuous on the interval [a,x] and that all derivatives up to the (n+1)th derivative exist and are continuous on the interval [a,x]. It also assumes that the polynomial used for approximation is the Taylor polynomial of degree n for the function at x=a.

5. How is the Lagrange remainder used in practice?

The Lagrange remainder is used in practice to determine the accuracy of polynomial approximations. For example, it can be used to estimate the error when calculating integrals or solving differential equations using numerical methods. It can also be used to determine the number of terms needed in a Taylor series to achieve a desired level of accuracy.

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