Problem 28 - Thomas Calculus

Question

Use power series operations to find the Taylor series at \(x=0\) for the functions in Exercises \(11-28 .\) $$\ln (1+x)-\ln (1-x)$$

Step-by-Step Solution

Answer

The Taylor series for \(\ln(1+x) - \ln(1-x)\) is \(\sum_{n=0}^{\infty} \frac{2x^{2n+1}}{2n+1}\).

Step 1: Identify the Basic Series

Recognize that the function \ln(1+x)\ is represented by the power series: \[\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^n}{n} = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots\] Similarly, for \ln(1-x)\, the power series is given by:\[\ln(1-x) = - (x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots )\]

Step 2: Subtract the Series

Subtract the series for \ln(1-x)\ from the series for \ln(1+x)\. Using the basic formulas from Step 1, calculate:\[\ln(1+x) - \ln(1-x) = \left( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \right) - \left( -x - \frac{x^2}{2} - \frac{x^3}{3} - \cdots \right)\]

Step 3: Simplify the Expression

Combine like terms from the subtraction:\[= \left( x + x \right) + \left( -\frac{x^2}{2} + \frac{x^2}{2} \right) + \left( \frac{x^3}{3} + \frac{x^3}{3} \right) + \cdots\]This simplifies to:\[2x + \frac{2x^3}{3} + \cdots\]

Step 4: Identify the Pattern

Notice that each term in the series has the form \ \frac{2x^{2n-1}}{2n-1} \ where \ n \ starts from 1. Thus, the Taylor series for \ \ln(1+x)-\ln(1-x) \ is:\[\sum_{n=1}^\infty \frac{2x^{2n-1}}{2n-1} = 2x + \frac{2x^3}{3} + \frac{2x^5}{5} + \cdots\]

Step 5: Finalize the Taylor Series

Write the complete Taylor series for \ln(1+x)-\ln(1-x)\:\[\ln(1+x) - \ln(1-x) = \sum_{n=0}^{\infty} \frac{2x^{2n+1}}{2n+1} \]

Key Concepts

Power Series

A power series is a way of expressing a function as an infinite sum of terms that are powers of a variable. It takes the form:

• \( a_0 + a_1x + a_2x^2 + a_3x^3 + \cdots \)

Here, \(a_n\) are constants called coefficients, and \(x\) is the variable. In the case of the natural logarithm functions, power series provide a tool for approximating these functions under certain conditions, particularly when \(-1 < x < 1\).
By substituting different values into the power series, we can get polynomial approximations that become more accurate with more terms.

Logarithmic Functions

Logarithmic functions, especially the natural logarithm \(\ln(x)\), play a crucial role in mathematics, particularly in calculus. They transform products into sums, which simplifies complex multiplication and division problems. The natural logarithm has the base \(e\), an irrational constant, approximately equal to 2.71828. For functions like \(\ln(1+x)\) or \(\ln(1-x)\), power series are useful as they convert the logarithmic function into an algebraic series.
This series form is especially handy for calculus operations like differentiation and integration.

Series Expansion

Series expansion involves expressing a function as a sum of simpler terms. In the context of the Taylor series, an expansion of \(\ln(1+x)\) is:

• \(\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \)

It represents the function as a sum of an infinite series of terms. The Taylor series is centered at \(x=0\), providing approximations that work best around this point. Such expansions are crucial for finding solutions to equations and creating approximations where direct computation might be complex.
This is the backbone of many approximation and analysis techniques.

Mathematical Series Operations

Mathematical series operations involve manipulating series to derive or simplify complex problems. In the given exercise, subtraction is the main operation. When two series are subtracted, like \(\ln(1+x) - \ln(1-x)\), you combine corresponding terms from each series. Here, every \(x\) term effectively adds up its like pair through subtraction, simplifying the expression. In the solution, this operation condenses the series into:

• \(\ln(1+x) - \ln(1-x) = 2x + \frac{2x^3}{3} + \cdots \)

Recognizing patterns in terms is another key operation. It allows one to predict the form of each term without listing them all explicitly. This capability is crucial for crafting generalized results or series like a Taylor series.