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Magazine Chapter 9: Problem 17
Use sigma notation to write the Taylor series about \(x=x_{0}\) for the function. $$ \ln x ; x_{0}=1 $$
Short Answer
Expert verified
\(\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}(x-1)^n\)
Step by step solution
01
Recall the Taylor Series Formula
The Taylor series expansion of a function \( f(x) \) about a point \( x = x_0 \) is given by the formula:\[f(x) = f(x_0) + f'(x_0)(x-x_0) + \frac{f''(x_0)}{2!}(x-x_0)^2 + \cdots + \frac{f^n(x_0)}{n!}(x-x_0)^n + \cdots\]This can be written in sigma notation as:\[f(x) = \sum_{n=0}^{\infty} \frac{f^n(x_0)}{n!}(x-x_0)^n\]where \( f^n(x_0) \) is the \( n^{th} \) derivative of \( f \) evaluated at \( x_0 \).
02
Identify the Function and Its Derivatives
The function given is \( \ln x \). We need to find the derivatives of \( \ln x \) and evaluate them at \( x_0 = 1 \). The derivatives are as follows:\[ f(x) = \ln x, \quad f'(x) = \frac{1}{x}, \quad f''(x) = -\frac{1}{x^2}, \quad f'''(x) = \frac{2}{x^3}, \quad \ldots \]Evaluated at \( x_0 = 1 \), these derivatives are:\[ f(1) = 0, \quad f'(1) = 1, \quad f''(1) = -1, \quad f'''(1) = 2, \quad \ldots \]This pattern continues where the \( n^{th} \) derivative at \( x_0 = 1 \) is \((-1)^{n-1}(n-1)!\).
03
Substitute in the Taylor Series Formula
Substitute the derivatives into the Taylor series formula:\[\ln x = \sum_{n=0}^{\infty} \frac{f^n(1)}{n!}(x-1)^n\]where:- \( f^0(1) = \ln 1 = 0 \), so the term corresponding to \( n=0 \) vanishes.- \( f^n(1) = (-1)^{n-1}(n-1)! \) for \( n \geq 1 \). Thus:\[\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}(n-1)!}{n!}(x-1)^n\]Simplifying the factorial part:\[\frac{(n-1)!}{n!} = \frac{1}{n}\]This reduces the series to:\[\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}(x-1)^n\]
04
Write the Final Answer in Sigma Notation
The Taylor series for \( \ln x \) about \( x_0 = 1 \) in sigma notation is:\[\ln x = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}(x-1)^n\]This represents the infinite series expansion of \( \ln x \) centered at \( x=1 \).
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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 powerful mathematical tool used to express the summation of a sequence of terms in a compact form. Instead of writing long sums, we use the Greek letter sigma (\( \Sigma \)) to represent summation.
- For instance, the sum of a sequence from \( n=1 \) to infinity is written as \( \sum_{n=1}^{\infty} a_n \), where \( a_n \) represents each term in the sequence.
- This notation not only simplifies complex series but also provides a clear format for defining and manipulating the terms mathematically.
Derivatives
Derivatives form the cornerstone of calculus and are essential in computing Taylor series. They represent the rate at which a function is changing at any point and are crucial for finding the series representation.
- The \( n^{th} \) derivative of a function \( f(x) \) is denoted as \( f^n(x) \), and it gives insight into the function's behavior at particular points, such as \( x_0 = 1 \) for our example function \( \ln x \).
- Each derivative provides a new term in the Taylor series, contributing to a more accurate approximation of the function around a specific point.
Natural Logarithm
The natural logarithm, denoted as \( \ln x \), is a fundamental function in mathematics especially relevant to calculus and exponential growth. This function provides the time needed to reach a certain level of growth under continuous compounding.
- Its base is the irrational number \( e \), approximately equal to 2.718, which makes the natural log unique compared to other logarithmic bases.
- It is the inverse function of the exponential function \( e^x \), meaning \( \ln(e^x) = x \) and \( e^{\ln x} = x \).
- At \( x_0 = 1 \), \( \ln 1 = 0 \), establishing the initial point of expansion.
- The natural logarithm is only defined for positive values of \( x \) but becomes particularly simple around \( x = 1 \), benefiting series expansion.
