Question

{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85

Accepted Answer

Identify the function to approximate: here, it is $$ f(x) = \ln(x) . We $$want to approximate $$ \ln(0.85) $$ using a Taylor polynomial centered at some point $$ a . $$Choose a center $$ a $$ close to 0.85 to ensure faster convergence and smaller error. A common choice is $$ a = 1 $$ because $$ \ln(1) = 0 $$ and derivatives of $$ \ln(x)  at 1 $$are easy to compute. Write the Taylor series expansion of $$ \ln(x) $$ about $$ a = 1 $$:

$$ \ln(x) = \ln(1) + \sum_{n=1}^{\infty} \frac{f^{(n)}(1)}{n!} (x - 1)^n $$

where $$ f^{(n)}(x) $$ denotes the $$ n -th $$derivative of $$ \ln(x) . $$Determine the general form of the $$ n -th $$derivative of $$ \ln(x)  at$$ $$ x=1 $$ and write the Taylor polynomial of order $$ N  as$$:

$$ P_N(x) = \sum_{n=1}^N \frac{(-1)^{n-1} (n-1)!}{n!} (x - 1)^n = \sum_{n=1}^N (-1)^{n-1} \frac{(x-1)^n}{n}

($$since the derivatives of $$ \ln(x)  at 1 $$follow a known pattern). Use the Lagrange remainder formula to bound the absolute error:

$$ |R_{N}(x)| = \left| \frac{f^{(N+1)}(c)}{(N+1)!} (x - 1)^{N+1} \right| \leq 10^{-3} $$

where $$ c  is $$some number between $$ x = 0.85 $$ and $$ a = 1 . $$Find the smallest $$ N $$ such that this inequality holds.

{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85

Verified video answer for a similar problem:

Key Concepts

Taylor Polynomial Approximation

Error Bound for Taylor Polynomials

Choice of Center in Taylor Series

{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)ln 0.85

Verified video answer for a similar problem:

Key Concepts

Taylor Polynomial Approximation

Error Bound for Taylor Polynomials

Choice of Center in Taylor Series

{Use of Tech} Number of terms What is the minimum order of the Taylor polynomial required to approximate the following quantities with an absolute error no greater than 10⁻³ ? (The answer depends on your choice of a center.)
ln 0.85