Problem 20 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Find the Maclaurin series for \(\ln (2+x)\) along with its radius of convergence.

Step-by-Step Solution

Answer

The Maclaurin series for \(\ln (2 + x)\) is given by the series \((-1)^{n-1} * x^n / n \) and its radius of convergence is 1.

Step 1: Reviewing the formula

To derive the Maclaurin series, recall the general formula for a Maclaurin series which is \(f(x) = f(0) + f'(0)x + f''(0)x^2/2! + f'''(0)x^3/3! + ... \). Note that for this problem, the series will start from \(n = 1\) because the function \(\ln(2+x)\) is undefined at \(x = 0\).

Step 2: Computing derivatives

Firstly, compute the derivative of the function at \(x = 0\) for a few terms to observe a pattern. The \(n^{th}\) derivative of \(\ln (2 + x)\) evaluated at x = 0 is \((-1)^{n-1} (n-1)! / 2^n \).

Step 3: Building the series

Now that the derivatives have been evaluated, the Maclaurin series for \(\ln (2 + x)\) is \((-1)^{n-1} * x^n / n \). The sign alternates due to the \((-1)^{n-1}\) term.

Step 4: Determining radius of convergence

The radius of convergence can be found using the Ratio Test, where the absolute value of the ratio of consecutive terms will be less than 1 within the range of convergence. This yields \( R = \(\lim_{n \to \infty}\) |a_{n+1} / a_n| = 1 \).

Key Concepts

Radius of Convergence

The radius of convergence is an important concept when dealing with power series, including Maclaurin and Taylor series. It tells us the interval within which the series will converge to the actual function.

• For a power series centered at a point, the radius of convergence, often denoted as \( R \), describes how far the series extends around that point before it diverges.
• The series converges for all \( x \) such that the distance from the center is less than \( R \).
• To determine \( R \) for the Maclaurin series of \( \ln(2 + x) \), the exercise uses the Ratio Test.

Using the Ratio Test, we find that the series converges when the ratio of the absolute values of consecutive terms is less than 1. In this case, the solution shows that the limit is \( 1 \), indicating a radius of convergence of \( 1 \). This means the series converges for \( |x| < 1 \).

Derivatives

Derivatives are central to forming Maclaurin series. For any function, we need to compute derivatives at a specific point to find the terms of its series.

• The Maclaurin series is essentially a Taylor series centered at \( x = 0 \).
• We compute derivatives of \( \ln(2+x) \) because the series construction requires them.
• Each term in the series corresponds to a derivative of a certain order, evaluated at \( x = 0 \).

For \( \ln(2+x) \), it is crucial to observe the pattern in the derivatives because it helps to construct the series. The \( n^{th} \) derivative evaluated at \( x = 0 \) is \((-1)^{n-1} (n-1)! / 2^n\). Noticing such patterns simplifies the process of building further terms.

Power Series

Understanding power series is fundamental when we're discussing Maclaurin series. A power series is an infinite series of the form \( \sum_{n=0}^{\infty} a_n (x-c)^n \), where \( c \) is the center.

• When \( c = 0 \), the series is a Maclaurin series.
• The coefficients \( a_n \) are derived from the function's derivatives at the center.
• Power series can represent many functions within the radius of convergence.

In our case, we're using a Maclaurin series for \( \ln(2+x) \), and the function representation becomes an infinite sum involving powers of \( x \). This provides a polynomial approximation of the natural logarithm function around the center (0 in Maclaurin's case).

Convergence Tests

Convergence tests help us determine if a series converges or diverges. The Ratio Test is one of the most common methods and is especially useful for power series.

• The Ratio Test compares the absolute values of terms \( a_n \) and \( a_{n+1} \) from a series.
• If \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1 \), the series converges.
• If the limit is greater than 1 or does not exist, the series diverges.

In our exercise, using the Ratio Test helps find that the series for \( \ln(2+x) \) has a radius of convergence \( R = 1 \). This means that within this interval, the series converges, accurately approximating the function.