Problem 62 - Thomas Calculus

Question

Assume that the series \(\sum a_{n} x^{n}\) converges for \(x=4\) and diverges for \(x=7 .\) Answer true \((T),\) false \((F),\) or not enough information given \((\mathrm{N})\) for the following statements about the series. $$ \begin{array}{l}{\text { a. Converges absolutely for } x=-4} \\ {\text { b. Diverges for } x=5} \\ {\text { c. Converges absolutely for } x=-8.5} \\\ {\text { d. Converges absolutely for } x=-8.5} \\ {\text { e. Diverges for } x=8} \\ {\text { f. Diverges absolutely for } x=0} \\ {\text { g. Converges absolutely for } x=-7.1}\end{array} $$

Step-by-Step Solution

Answer

a. T; b. F; c. F; d. F; e. T; f. F; g. F.

Step 1: Understand the problem

The problem provides information about the convergence and divergence of a power series at specific points and asks for answering whether other statements about the series are true, false, or not enough information given.

Step 2: Determine the radius of convergence

The series \(\sum a_n x^n\) converges for \(x = 4\) and diverges for \(x = 7\). This indicates the convergence interval is from \(-R, R\) including \(x=4\) but excluding \(x=7\). Therefore, \(R\) should satisfy \(4 \leq R < 7\). We assume \(R = 4\) as it's the smallest possible value that satisfies the given conditions.

Step 3: Analyze Statement a

Given \(R=4\), the series converges absolutely for the interval \([-R, R]\), which means it converges absolutely for \(x=-4\). Thus, the statement a is true (T).

Step 4: Analyze Statement b

For the series to converge, \(|x|

Step 5: Analyze Statement c

For absolute convergence \(|-8.5| = 8.5\) must be less than \(R\). Since \(R=4\), and \(8.5\) is more than \(4\), the series does not converge absolutely for \(x=-8.5\). Statement c is false (F).

Step 6: Analyze Statement d

This statement is identical to c. Thus, the same logic applies, and statement d is false (F).

Step 7: Analyze Statement e

For divergence, \(|8| = 8\) must be outside the interval of convergence \([-4, 4]\). The series will diverge for \(x=8\), confirming statement e true (T).

Step 8: Analyze Statement f

For \(x=0\), \(|0| = 0\) is within any radius of convergence, therefore \(x=0\) converges absolutely, making statement f false (F).

Step 9: Analyze Statement g

\(|-7.1| = 7.1\) is more than \(R=4\), meaning \(x=-7.1\) is outside the convergence radius. Statement g is false (F).

Key Concepts

Power Series

A power series is a series of the form \( \sum a_n x^n \), where each term is a product of a coefficient \( a_n \) and a power of \( x \). Power series are similar to polynomials, but they can have infinitely many terms. They are used to represent functions across a range. Understanding power series involves analyzing when and where they converge. Convergence means that as more terms are added, the series approaches a certain value.
Power series are encountered frequently in calculus and complex analysis because they provide a way to express complex functions as sums of simpler ones. They are the basis of Taylor and Maclaurin series, which approximate functions using derivatives.

• The key part of a power series is the coefficients \( a_n \) which can dictate how fast the terms grow or shrink.
• The variable \( x \) influences whether the series converges at certain values or not.

Learning how to determine the convergence of a power series involves understanding the Radius of Convergence.

Radius of Convergence

The radius of convergence \( R \) is the distance from the center \( c \) of a power series to the "edge" of convergence on the number line. It defines the interval where the power series converges. In simpler terms, it tells us what values of \( x \) will make the series approach a certain value.
The radius can be found using the formula derived from the ratio test or the root test. If the series converges for \( x = 4 \) but diverges for \( x = 7 \), then the radius of convergence will be somewhere between these points.

• With \( R = 4 \), the series converges for values of \( x \) within \( -4 \) and \( 4 \).
• The interval for absolute convergence will stretch to both positive and negative sides equally, centered at the origin unless otherwise specified.

Understanding this helps in predicting the behavior of the power series across different values of \( x \).

Absolute Convergence

Absolute convergence refers to convergence of the series even when the terms are replaced by their absolute values. Essentially, a series \( \sum a_n \) converges absolutely if \( \sum |a_n| \) is also convergent. This is a stronger condition than regular convergence. If a series is absolutely convergent, then it is also convergent, but the converses may not be true.
For example, when considering tertiary locations like \( x = -4 \), \( x = 5 \), or \( x = -8.5 \), the absolute convergence condition must still hold.

• If \( |x| < R \), then the series will absolutely converge at \( x \) points.
• If the series is not absolutely convergent at \( x \), then it suggests either divergence or conditional convergence.

Analyzing series for absolute convergence involves examining these absolute value conditions for various \( x \) within the allowable range.

Divergence

A series diverges if it does not approach a finite limit as more terms are added. In simple terms, it means the series does not settle at any specific value no matter how many terms you calculate. In terms of our power series, points outside of the radius of convergence typically lead to divergence.
For the exercise, this was seen at locations like \( x = 7 \) and tested at \( x = 5 \) or \( x = 8 \).

• Divergence often implies that the value "jumps around" without nearing any point.
• Sometimes, a series may diverge even though the terms themselves do not grow much. This can occur with alternating terms.

If divergence occurs at a certain point, that point lies outside the radius of convergence, which dictates where the series no longer converges.

Interval of Convergence

The interval of convergence is the set of \( x \) values for which the power series converges. It includes all the points where the series approaches a finite limit as described by the radius of convergence. Inside this interval, the series will always converge to a value, while outside, there will be divergence.
In the exercise, convergence was suggested for \( x = 4 \), marking the series' behavior range, excluding \( x = 7 \).

• The interval typically looks like \([-R, R]\), possibly with endpoints included if there is boundary convergence.
• Sometimes certain endpoints must be tested separately, as convergence at the endpoints may differ from that inside the interval.

The interval of convergence is crucial for determining where the series will be reliable and applicable for function evaluation.

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Problem 62

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Key Concepts

Power Series

A power series is a series of the form \( \sum a_n x^n \), where each term is a product of a coefficient \( a_n \) and a power of \( x \). Power series are similar to polynomials, but they can have infinitely many terms. They are used to represent functions across a range. Understanding power series involves analyzing when and where they converge. Convergence means that as more terms are added, the series approaches a certain value.
Power series are encountered frequently in calculus and complex analysis because they provide a way to express complex functions as sums of simpler ones. They are the basis of Taylor and Maclaurin series, which approximate functions using derivatives.

• The key part of a power series is the coefficients \( a_n \) which can dictate how fast the terms grow or shrink.
• The variable \( x \) influences whether the series converges at certain values or not.

Learning how to determine the convergence of a power series involves understanding the Radius of Convergence.

Radius of Convergence

The radius of convergence \( R \) is the distance from the center \( c \) of a power series to the "edge" of convergence on the number line. It defines the interval where the power series converges. In simpler terms, it tells us what values of \( x \) will make the series approach a certain value.
The radius can be found using the formula derived from the ratio test or the root test. If the series converges for \( x = 4 \) but diverges for \( x = 7 \), then the radius of convergence will be somewhere between these points.

• With \( R = 4 \), the series converges for values of \( x \) within \( -4 \) and \( 4 \).
• The interval for absolute convergence will stretch to both positive and negative sides equally, centered at the origin unless otherwise specified.

Understanding this helps in predicting the behavior of the power series across different values of \( x \).

Absolute Convergence

Absolute convergence refers to convergence of the series even when the terms are replaced by their absolute values. Essentially, a series \( \sum a_n \) converges absolutely if \( \sum |a_n| \) is also convergent. This is a stronger condition than regular convergence. If a series is absolutely convergent, then it is also convergent, but the converses may not be true.
For example, when considering tertiary locations like \( x = -4 \), \( x = 5 \), or \( x = -8.5 \), the absolute convergence condition must still hold.

• If \( |x| < R \), then the series will absolutely converge at \( x \) points.
• If the series is not absolutely convergent at \( x \), then it suggests either divergence or conditional convergence.

Analyzing series for absolute convergence involves examining these absolute value conditions for various \( x \) within the allowable range.

Divergence

A series diverges if it does not approach a finite limit as more terms are added. In simple terms, it means the series does not settle at any specific value no matter how many terms you calculate. In terms of our power series, points outside of the radius of convergence typically lead to divergence.
For the exercise, this was seen at locations like \( x = 7 \) and tested at \( x = 5 \) or \( x = 8 \).

• Divergence often implies that the value "jumps around" without nearing any point.
• Sometimes, a series may diverge even though the terms themselves do not grow much. This can occur with alternating terms.

If divergence occurs at a certain point, that point lies outside the radius of convergence, which dictates where the series no longer converges.

Interval of Convergence

The interval of convergence is the set of \( x \) values for which the power series converges. It includes all the points where the series approaches a finite limit as described by the radius of convergence. Inside this interval, the series will always converge to a value, while outside, there will be divergence.
In the exercise, convergence was suggested for \( x = 4 \), marking the series' behavior range, excluding \( x = 7 \).

• The interval typically looks like \([-R, R]\), possibly with endpoints included if there is boundary convergence.
• Sometimes certain endpoints must be tested separately, as convergence at the endpoints may differ from that inside the interval.

The interval of convergence is crucial for determining where the series will be reliable and applicable for function evaluation.