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Magazine Chapter 9: Problem 18
Use sigma notation to write the Taylor series about \(x=x_{0}\) for the function. $$ \ln x ; x_{0}=e $$
Short Answer
Expert verified
\(\ln(x) = 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \left(\frac{x-e}{e}\right)^n.\)
Step by step solution
01
Understand the Function and Taylor Series Definition
The function given is the natural logarithm function \(\ln(x)\). A Taylor series is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. The formula for a Taylor series about \(x=a\) is: \(\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n.\)
02
Find the Derivatives at the Point
First, calculate the derivatives of \(\ln(x)\) and evaluate them at \(x_0 = e\). - \(f(x) = \ln(x) \rightarrow f(e) = \ln(e) = 1\)- First derivative: \(f'(x) = \frac{1}{x} \rightarrow f'(e) = \frac{1}{e}\)- Second derivative: \(f''(x) = -\frac{1}{x^2} \rightarrow f''(e) = -\frac{1}{e^2}\)- Third derivative: \(f'''(x) = \frac{2}{x^3} \rightarrow f'''(e) = \frac{2}{e^3}\), and so on.
03
Express the General Term
For the general term in the Taylor series, apply the formula: \(\frac{f^{(n)}(e)}{n!} (x-e)^n.\)From the derivatives, notice: \(f^{(n)}(e) = (-1)^{n+1}\frac{(n-1)!}{e^n}\) for \(n \geq 1\). The logarithm series does not have a defined 0th degree term except for the offset, which is \(f(e) = 1\).
04
Set Up the Sigma Notation
Using the general term from Step 3, the Taylor series can be set up in sigma notation:\[1 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}\cdot(x-e)^n}{n\cdot e^n}.\]Note that the constant term comes from \(\ln(e) = 1\).
05
Write the Final Series Expression
The Taylor series using sigma notation is given by:\[\ln(x) \approx 1 + \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \left(\frac{x-e}{e}\right)^n.\]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sigma Notation
Sigma notation is a compact and powerful way to represent sums, especially when dealing with infinite series like Taylor series. It uses the Greek letter sigma (鈭�) to indicate a sum and provides a concise way to express long sums without writing each term individually. In the Taylor series, sigma notation helps us express an infinite series that approximates a function by combining its derivatives.
- Basic Structure: Sigma notation is written as \( \sum_{n=0}^{\infty} a_n \), where \( n \) represents the index of summation, \( a_n \) is the general term in the sequence, and the limits of summation indicate where \( n \) starts and ends.
- Application to Taylor Series: In the context of the Taylor series, the general term \( \frac{f^{(n)}(a)}{n!} (x-a)^n \) is repeated indefinitely, captured succinctly by sigma notation.
Natural Logarithm
The natural logarithm, denoted as \( \ln(x) \), is the inverse function of the exponential function \( e^x \). It plays a critical role in calculus, especially in the context of derivatives and series expansions.
- Base of Natural Logarithms: The natural logarithm uses the constant \( e \) (approximately 2.71828) as its base, which arises naturally in the study of growth and decay processes.
- Simplification at \( e \): When evaluated at \( x = e \), the natural logarithm simplifies to \( \ln(e) = 1 \), which forms the constant term in its Taylor series expansion.
- Usefulness in Calculus: Natural logarithms are used in integration, solving certain differential equations, and in modeling exponential growth/decay.
Derivatives
Derivatives provide fundamental insights into how functions change. In the context of Taylor series, derivatives are used to construct an approximation of a function around a specific point, described by an infinite series.
- Role in Taylor Series: Derivatives determine each term in the Taylor series expansion. Calculating the nth derivative of a function, evaluated at a given point, is essential for forming every term in the series.
- Successive Derivatives: For \( \ln(x) \), the derivatives follow a recognizable pattern: first is \( 1/x \), second is \( -1/x^2 \), third is \( 2/x^3 \), and so on. This pattern allows for easier computation of terms in the series.
- Evaluating at a Point: To use derivatives in a Taylor series, calculate them at the chosen point around which the series is centered. For example, evaluating derivatives of \( \ln(x) \) at \( x = e \) provides the necessary coefficients for the Taylor series of \( \ln(x) \).
