Problem 23 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 9-28, find the convergence set for the given power series. Hint: First find a formula for the nth term; then use the Absolute Ratio Test. $$ \frac{x-1}{1}+\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}+\frac{(x-1)^{4}}{4}+\cdots $$

Step-by-Step Solution

Answer

The series converges when \( |x-1| < 1 \).

Step 1: Identify the Power Series

The given series can be rewritten in a more familiar form: \( \sum_{n=1}^{\infty} \frac{(x-1)^n}{n} \). Here, \( a_n = \frac{(x-1)^n}{n} \) represents the general term of the series.

Step 2: Applying the Absolute Ratio Test

The Absolute Ratio Test is used to determine the convergence of a power series. We calculate the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). For our series, this gives: \( L = \lim_{n \to \infty} \left| \frac{(x-1)^{n+1}/(n+1)}{(x-1)^n/n} \right| \). Simplifying the ratio, \( L = \lim_{n \to \infty} \left| \frac{(x-1)^{n+1}n}{(x-1)^n(n+1)} \right| = \lim_{n \to \infty} \left| \frac{(x-1)n}{n+1} \right| \).

Key Concepts

nth term formula

When working with power series, identifying a formula for the nth term is essential. Imagine a train with infinite cars, each car representing a term in the series. To see how the series behaves, we first need to come up with an expression that represents any car, or term, in the train. This is known as the nth term formula.
In the problem given, our series is written as:

• \( \frac{x-1}{1}+\frac{(x-1)^{2}}{2}+\cdots \)
• Or more compactly, \( \sum_{n=1}^{\infty} \frac{(x-1)^n}{n} \)

Understanding this formula is like setting a pattern for each term. Here, it tells us how each term grows based on \( (x-1) \) and \( n \). This insight is the first step in analyzing any series and determining how and where it converges.

Absolute Ratio Test

The Absolute Ratio Test is a powerful tool for analyzing the convergence of power series. Think of it as a detective investigating the behavior of the series.
For the series \( a_n \), it involves calculating the ratio of successive terms. In our situation:

• Compare \( a_{n+1} = \frac{(x-1)^{n+1}}{n+1} \) with \( a_n = \frac{(x-1)^n}{n} \).

This gives us the ratio:

• \( \frac{a_{n+1}}{a_n} = \frac{(x-1)^{n+1}/(n+1)}{(x-1)^n/n} = \frac{(x-1)n}{n+1} \).

Simplifying further helps us check the limit as \( n \to \infty \). If the limit is less than 1, the series converges for those \( x \) values, and this helps pinpoint the series' convergence range.

convergence set

Once we've applied the Absolute Ratio Test, we turn our attention to the convergence set. This set tells us the x-values where the power series will converge. Just as plants only grow in suitable conditions, a series only converges for specific x-values.
The limit previously calculated aids in determining this set:

• Find \( L = \lim_{n \to \infty} \left| \frac{(x-1)n}{n+1} \right| \).
• As \( n \to \infty \), the ratio approaches \( |x-1| \).

This ratio is crucial because the series converges when \( |x-1| < 1 \). This inequality reveals the interval of convergence, which is the set \( 0 < x < 2 \).

mathematical series analysis

Analyzing mathematical series requires tools from calculus and algebra to understand their behavior. Imagine series analysis as a map that guides you through the landscape of convergence and divergence.
The task includes several steps:

• Identify the nth term formula to set the foundation.
• Apply tests like the Absolute Ratio Test to determine dynamics.
• Find the convergence set, specifying where the series hugs close to a certain path.
  

Each component builds upon the last, ultimately helping us understand how the series behaves across different x-values. This systematic approach ensures that we don't get lost amidst the potentially infinite terms, offering a clear vision of convergence.