Solve the problem.
-The series $\ln ( \mathrm { x } ) = ( \mathrm { x } - 1 ) - \frac { ( \mathrm { x } - 1 ) ^ { 2 } } { 2 } + \frac { ( \mathrm { x } - 1 ) ^ { 3 } } { 3 } - \ldots + ( - 1 ) ^ { \mathrm { n } - 1 } \frac { ( \mathrm { x } - 1 ) ^ { \mathrm { n } } } { \mathrm { n } }$ can be used to estimate the value of $\ln ( a )$ for any positive real number a such that $0

Question

Solve the problem.
-The series $\ln ( \mathrm { x } ) = ( \mathrm { x } - 1 ) - \frac { ( \mathrm { x } - 1 ) ^ { 2 } } { 2 } + \frac { ( \mathrm { x } - 1 ) ^ { 3 } } { 3 } - \ldots + ( - 1 ) ^ { \mathrm { n } - 1 } \frac { ( \mathrm { x } - 1 ) ^ { \mathrm { n } } } { \mathrm { n } }$ can be used to estimate the value of $\ln ( a )$ for any positive real number a such that $0 < a \leq 2$. Use the first five terms of this series to estimate $\ln ( 1.54 )$. Round to the nearest thousandth.
A) $- 0.495$
B) $0.435$
C) $1.995$
D) $1.325$

Quizplus · Accepted Answer

The series given is the Taylor series expansion of $\ln(x)$ around $x=1$. To estimate $\ln(1.54)$, substitute $x=1.54$ into the series and use the first five terms: $\ln(1.54) \approx (1.54 - 1) - \frac{(1.54 - 1)^2}{2} + \frac{(1.54 - 1)^3}{3} - \frac{(1.54 - 1)^4}{4} + \frac{(1.54 - 1)^5}{5}$. Calculating this gives a value close to $0.435$, rounded to the nearest thousandth.

Solve the Problem Can Be Used to Estimate the Value Of