Problem 34 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(\sum_{n=0}^{\infty} a_{n}(x-3)^{n}\) converges at \(x=-1\). Why can you conclude that it converges at \(x=6 ?\) Can you be sure that it converges at \(x=7\) ? Explain.

Step-by-Step Solution

Answer

Converges at x=6 because it's within the radius. Convergence at x=7 is uncertain.

Step 1: Understanding Power Series Convergence

A power series \( \sum_{n=0}^{\infty} a_{n}(x-c)^{n} \) converges for values of \( x \) within a certain distance from the center \( c \), known as the radius of convergence \( R \). This means that if the series converges at a point \( x = a \), then it also converges for all \( x \) such that \(|x - c| < R\).

Step 2: Assess Given Convergence Information

The series \( \sum_{n=0}^{\infty} a_{n}(x-3)^{n} \) is known to converge at \( x = -1 \). Since the center of the series is \( 3 \), this implies that convergence occurs when \(|-1 - 3|\), which equals 4, is within the radius of convergence.

Step 3: Determine Convergence at x = 6

To determine if the series converges at \( x = 6 \), calculate \(|6 - 3| = 3\). Since \( 3 < 4 \), where 4 is the distance for known convergence at \(x = -1\), the series converges at \( x = 6 \) as well because 6 is closer to 3 than -1 is.

Step 4: Evaluate Convergence at x = 7

Check convergence at \( x = 7 \) by calculating \(|7 - 3| = 4\). This equals the distance we calculated for convergence at \( x = -1 \). While this boundary value indicates the series may converge, it is not guaranteed since boundary convergence depends on specific series behavior.

Key Concepts

Radius of Convergence

The radius of convergence is a key concept that helps determine where a power series converges. Think of it as the distance you can move from the center of the series and still have it converge. For a power series of the form \[\sum_{n=0}^{\infty} a_{n}(x-c)^{n},\]this radius, denoted by \( R \), provides a guideline. The series will converge for any value \( x \) as long as \[|x - c| < R.\]This means you take the center \( c \) of the series and can "travel" a distance \( R \) in either direction while maintaining convergence.

• If you know the series converges at some point \( x = a \), you've essentially found one boundary of \( R \).
• Within this radius, you can be sure the series is converging smoothly.

To calculate the radius, techniques like the Ratio Test or Cauchy-Hadamard Theorem can be used.

Center of Series

The center of a power series is a fixed value around which the series "blooms" or expands. In the example \( \sum_{n=0}^{\infty} a_{n}(x-3)^{n} \), the center is \( c = 3 \). This is the point from which you measure outwards when considering convergence.

• The center is crucial since it sets the starting point for measuring the convergence radius.
• All distance measurements in regard to convergence start from this center.

Choosing different values for \( c \) adjusts the series alignment and location. However, it does not change its convergence properties. Think of the center as the "home base" of the series from which all convergence is evaluated.

Boundary Value Convergence

When you reach the edge of the radius of convergence, things become uncertain. Boundary value convergence refers to points on the very circle of convergence where \[|x - c| = R.\]At these boundary points, we cannot be automatically sure of convergence. Each point needs its own investigation.

• Unlike the areas well within the radius, boundaries require specific testing or knowledge of the series.
• The series might converge at some boundary points and diverge at others.

This characteristic of boundary behavior makes convergence analysis complex and requires extra testing methods.

Absolute Convergence

A series is said to have absolute convergence if it converges even when all its terms are made positive. For a power series,\[\sum_{n=0}^{\infty} |a_{n}(x-c)^{n}|\]converging implies it joins up neatly without any issues of fluctuation or spread, no matter the sign of terms. This is a strong form of convergence.

• Absolute convergence within the radius implies convergence at every point over that distance.
• It ensures stability across all possible values, not just for positive terms.

Thus, it gives greater assurance than conditional convergence, where a series converges only due to the specific balance of positive and negative parts.