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

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SUMMARY

The discussion focuses on bounding the error in Taylor series approximations for the function f(x) = ln(1+x) using the third-order Taylor polynomial P3(x). The Taylor series expansion provided is x - x²/2! + (2x³)/3!. The error is determined by the remainder term of the Taylor series, specifically the fourth-order polynomial, expressed as |R4(x)| = |f⁽⁴⁾(c)|/4! * |x|⁴, where c is in the interval [-1/2, 1/2]. The key to solving the problem lies in evaluating the fourth derivative of f(x) and finding its maximum absolute value within the specified interval.

PREREQUISITES
  • Understanding of Taylor series and polynomial approximations
  • Knowledge of derivatives and their applications in error analysis
  • Familiarity with the function f(x) = ln(1+x)
  • Basic calculus concepts, including limits and continuity
NEXT STEPS
  • Calculate the fourth derivative of f(x) = ln(1+x) and evaluate it over the interval [-1/2, 1/2]
  • Study the concept of Taylor series remainders and their significance in approximation
  • Explore error bounds in polynomial approximations using higher-order derivatives
  • Learn about convergence and divergence of Taylor series in different intervals
USEFUL FOR

Students studying calculus, mathematicians interested in approximation theory, and educators teaching Taylor series and error analysis.

francisg3
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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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Your formula for the remainder after n = 3 is

\frac{f^{(4)}(c)}{4!}(x-a)^4

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

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