Problem 23 - Calculus: Early Transcendentals

Question

Find a power series representation for \( f, \) and graph \( f \) and several partial sums \( s_n(x) \) on the same screen. What happens as \( n \) increases? \( f(x) = \ln \left( \frac {1 + x}{1 - x} \right) \)

Step-by-Step Solution

Answer

As the number of terms (\(n\)) increases, the partial sum better approximates \(f(x)\) within its radius of convergence.

Step 1: Express the Function Using Known Series

The function given is \( f(x) = \ln \left( \frac {1 + x}{1 - x} \right) \). We can utilize the known series expansion for the natural logarithm function: if \( \ln(1 + u) \) is expressed as \( u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \), then we can substitute \( u = \frac{2x}{1-x} \) (from \( \ln \left( \frac{1+x}{1-x} \right) = 2\ln(1+x) - 2\ln(1-x) \)) to form the series for our function.

Step 2: Find the Power Series for Each Log Term

Start by finding the power series for \( \ln(1 + x) \) as \( x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \) and \( \ln(1 - x) \) as \( -(x + \frac{x^2}{2} + \frac{x^3}{3} + \cdots) \). Thus, \( \ln \left( \frac{1+x}{1-x} \right) \) becomes \(2(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots) \).

Step 3: Combine the Series

The function can now be expressed as:\[ f(x) = 2\Big(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots\Big) = 2\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1}. \] This is the power series representation for \( f(x) \).

Step 4: Graph the Function and Partial Sums

Generate plots for \( f(x) \) and the partial sums, defined as \( s_n(x) = 2\Big(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots + \frac{x^{2n+1}}{2n+1}\Big) \). Use a tool like Desmos or Python (matplotlib) to visualize these. Plot several partial sums \( s_1(x), s_3(x), \) and \( s_5(x) \) along with the exact \( f(x) \).

Step 5: Analyze the Behavior as n Increases

Observe the graphs from Step 4 to see that as \( n \) increases, the partial sum \( s_n(x) \) becomes a better approximation of the function \( f(x) \). This is because a power series converges to the function it represents within its radius of convergence. The approximation improves as more terms are added.

Key Concepts

Natural Logarithm Function

The natural logarithm function, denoted as \( \ln(x) \), is a fundamental logarithmic function with the base of the transcendental number \( e \), approximately equal to 2.71828. This function is defined for all positive real numbers and has several key properties:

• Derivative: The derivative of \( \ln(x) \) is \( \frac{1}{x} \), which means it has a smooth curve continuously increasing, but at a decreasing rate.
• Inverse Function: The natural logarithm is the inverse of the exponential function \( e^x \), such that if \( y = e^x \), then \( x = \ln(y) \).
• Domain and Range: It is defined for \( x > 0 \), and its range is all real numbers.

In the context of power series, the natural logarithm's series expansion plays a crucial role. For an expression \( \ln(1+u) \), it can be expanded into the power series \( u - \frac{u^2}{2} + \frac{u^3}{3} - \cdots \). This expansion allows us to manipulate complex logarithmic expressions into more manageable components, which becomes especially helpful in series representation of more complex logarithmic functions.

Partial Sums

Partial sums are the sums of the first \( n \) terms of a sequence. In the context of series, a partial sum is used to approximate the total sum of the series. By adding more terms, the partial sum converges to the actual sum if the series itself converges.To see this in action, consider a power series representation of a function, such as the representation for \( f(x) = \ln \left( \frac{1+x}{1-x} \right) \). The power series is written as \(2(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots)\). Each partial sum is essentially a polynomial approximating the function:

• \( s_1(x) = 2(x) \) approximates \( f(x) \) in its simplest linear form.
• \( s_3(x) = 2(x + \frac{x^3}{3}) \) adds a cubic term to improve the approximation.
• \( s_5(x) = 2(x + \frac{x^3}{3} + \frac{x^5}{5}) \) further enhances accuracy with a fifth-degree term.

Observing partial sums through plotting provides insights into how closely these sums approximate the actual function \( f(x) \) as more terms get included.

Series Expansion

A series expansion is a way of representing a function as a series or sum of terms calculated from the values of its derivatives at a particular point. This process breaks down complex functions into infinite sums that can be handled more easily.For the function \( f(x) = \ln \left( \frac{1+x}{1-x} \right) \), the strategy involves finding the series representation of the simpler functions \( \ln(1+x) \) and \( \ln(1-x) \), which can be expressed as:

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

By combining these series with appropriate coefficients, we derive the series for the function \( \ln \left( \frac{1+x}{1-x} \right) \) as \( 2(x + \frac{x^3}{3} + \frac{x^5}{5} + \cdots) \). This illustrates how a more complex natural logarithm function can be represented efficiently as a power series, enabling easier analysis and computation of its properties.

Convergence of Series

Convergence of a series is a critical concept in determining whether a series representation accurately approximates a function as more terms are added. A series converges if the sequence of its partial sums approaches a finite limit.For power series, convergence depends on the value of \( x \) and the series itself. In our case, the series \( 2\sum_{n=0}^{\infty} \frac{x^{2n+1}}{2n+1} \) converges for \( |x| < 1 \). This ensures that, within this domain, adding more terms results in a series that closely approximates the function \( f(x) \).When analyzing convergence:

• Check the interval of convergence: Use techniques like the ratio test to determine where the series converges.
• Observe the behavior of partial sums: Plotting partial sums can give a visual confirmation of convergence, as they should increasingly resemble the function \( f(x) \).

Understanding convergence is essential for effectively using series expansion as it assures us that our polynomial approximation becomes ever closer to the true function, providing accurate computations within the specified range.