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Magazine Chapter 6: Problem 4
Using Taylor's Theorem, show that, for all \(x>0\),
$$
x-\frac{1}{2} x^{2}<\log (1+x)
Short Answer
Expert verified
The inequality follows from the Taylor series approximation of \( \log(1+x) \).
Step by step solution
01
Recall Taylor's Theorem
Taylor's Theorem states that for a function \( f \) that is \( n \)-times differentiable at a point \( a \), the function can be approximated near \( a \) as follows:\[f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots + \frac{f^{(n)}(a)}{n!}(x-a)^n + R_n(x)\]where \( R_n(x) \) is the remainder term. For \( f(x) = \log(1+x) \) and expanding about \( a = 0 \), this becomes important when comparing the approximations.
02
Differentiate and Find Terms
Compute the relevant derivatives of \( f(x) = \log(1+x) \):1. \( f(x) = \log(1+x) \)2. \( f'(x) = \frac{1}{1+x} \)3. \( f''(x) = -\frac{1}{(1+x)^2} \)4. \( f'''(x) = \frac{2}{(1+x)^3} \)These derivatives will help construct the Taylor series at \( x = 0 \).
03
Calculate the Taylor Series
Using the derivatives, the Taylor series for \( \log(1+x) \) at \( x = 0 \) up to the third order is:\[\log(1+x) = 0 + x - \frac{1}{2}x^2 + \frac{1}{3}x^3 + R_3(x)\]Where \( R_3(x) \) is the remainder term for the third-order approximation.
04
Evaluate the Remainder Term
The remainder term for the third-order Taylor polynomial is:\[R_3(x) = \frac{f^{(4)}(c)}{4!}x^4 = \frac{-6}{(1+c)^4}\frac{x^4}{4!}\]for some \( c \) between 0 and \( x \). Since \( x > 0 \) and \( (1+c)^4 > 1\), the impact of \( R_3(x) \) is small.
05
Establish the Inequality
Using the approximation and noting that the remainder terms are small for small values of \( x \):1. **Lower Bound:** - The Taylor series without the third and higher-order terms gives: \( x - \frac{1}{2}x^2 \). - This is less than \( \log(1+x) \) because remainder terms are positive.2. **Upper Bound:** - Including the third-order term: \( x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \). - This is greater than \( \log(1+x) \), accounting the possible negative higher-order remainders.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Logarithmic Approximation
Logarithmic approximation involves expressing \( \log(1+x) \) as a series expansion to simplify calculations. This method relies on Taylor's Theorem, which allows us to approximate functions with polynomials.
\( \log(1+x) \) is especially useful in mathematical analysis and various applications because it can be complex to handle directly.
The goal is to approximate \( \log(1+x) \) using Taylor's series about \( x = 0 \), transforming it into a polynomial form:
\( \log(1+x) \) is especially useful in mathematical analysis and various applications because it can be complex to handle directly.
The goal is to approximate \( \log(1+x) \) using Taylor's series about \( x = 0 \), transforming it into a polynomial form:
- Zero order: \( \log(1) = 0 \)
- First order: \( x \)
- Second order: \( -\frac{1}{2}x^2 \)
- Third order: \( +\frac{1}{3}x^3 \)
Derivatives
In mathematical analysis, finding derivatives helps us decipher the behavior of functions. For \( f(x) = \log(1+x) \), derivatives reveal essential components of its behavior as a polynomial.
The first few derivatives are:
The first few derivatives are:
- \( f'(x) = \frac{1}{1+x} \) represents the slope or rate of change.
- \( f''(x) = -\frac{1}{(1+x)^2} \) tells us about concavity, or how the function curves.
- \( f'''(x) = \frac{2}{(1+x)^3} \) continues to reveal subtle changes in curvature.
Remainder Term
The remainder term \( R_n(x) \) in Taylor's Theorem is critical for assessing the accuracy of an approximation. When approximating \( \log(1+x) \), the remainder indicates how much error might be present in the polynomial representation.
For our specific approximation up to the third degree:
Since \( x > 0 \) and \( (1+c)^4 > 1 \), the remainder \( R_3(x) \) is usually small for a small \( x \). This means the approximation \( x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \) is quite close to \( \log(1+x) \). Understanding this term helps gauge the precision of the approximation and ensures we know the bounds of our estimate.
For our specific approximation up to the third degree:
- \( R_3(x) = \frac{f^{(4)}(c)}{4!}x^4 \).
Since \( x > 0 \) and \( (1+c)^4 > 1 \), the remainder \( R_3(x) \) is usually small for a small \( x \). This means the approximation \( x - \frac{1}{2}x^2 + \frac{1}{3}x^3 \) is quite close to \( \log(1+x) \). Understanding this term helps gauge the precision of the approximation and ensures we know the bounds of our estimate.
