Problem 20 - Thomas Calculus

Question

In Exercises \(1-36,\) (a) find the series' radius and interval of convergence. For what values of \(x\) does the series converge (b) absolutely (c) conditionally? $$ \sum_{n=1}^{\infty} \sqrt[n]{n}(2 x+5)^{n} $$

Step-by-Step Solution

Answer

The series converges absolutely for \(-3 < x < -2\) and does not converge conditionally at the endpoints.

Step 1: Identify the Type of Series

The given series is \( \sum_{n=1}^{\infty} \sqrt[n]{n}(2x+5)^n \). We can identify this as a power series in \((2x+5)\). To determine its convergence properties, we need to find the radius and interval of convergence.

Step 2: Apply the Root Test

For a power series \( \sum a_n (x-c)^n \), the Root Test can be used to find the radius of convergence. Here, \( a_n = \sqrt[n]{n} (2x + 5)^n \). Thus, \[\lim_{n \to \infty} \left| a_n \right|^{1/n} = \lim_{n \to \infty} \left( \sqrt[n]{n}\right)^{1/n} \cdot \left|(2x + 5)\right| = \lim_{n \to \infty} n^{1/n^2} \cdot |2x + 5| = |2x + 5|.\]For convergence, \[\big| 2x + 5 \big| < 1\] which solves to the interval.

Step 3: Solve the Inequality for the Interval of Convergence

Solving \( |2x + 5| < 1 \), we break it into:1. \( 2x + 5 < 1 \) which simplifies to \( 2x < -4 \) and hence \( x < -2 \).2. \( 2x + 5 > -1 \) which simplifies to \( 2x > -6 \) and hence \( x > -3 \).So, the interval of convergence is \(-3 < x < -2\).

Step 4: Determine Absolute and Conditional Convergence

Since the root test yielded a strict inequality, the series converges absolutely within the open interval \(-3 < x < -2\). At the endpoints x = -3 and x = -2, we should check separately for absolute and conditional convergence, but as \( |2x + 5| \) becomes \(1\), the series neither absolutely nor conditionally converges at the endpoints.

Key Concepts

Radius of Convergence

When working with power series, the concept of the radius of convergence is essential. It tells us how far we can extend the variable, denoted here as \( x \), from the center of the series and still have the series converge.
For the series \( \sum_{n=1}^{\infty} \sqrt[n]{n}(2x+5)^n \), the radius of convergence can be found using the root test.Let's break it down:

• Identify the series as a power series \( \sum a_n (x-c)^n \). For this series, the \( a_n \) term is \( \sqrt[n]{n} \underline{\phantom{xxx}} (2x+5)^n \).
• Apply the root test: \[ \lim_{n \to \infty} \left| a_n \right|^{1/n} = |2x + 5| \] This simplifies directly to the expression \(|2x + 5| \), suggesting that for convergence we need \(|2x + 5| < 1 \).

Thus, the radius of convergence, \( R \), is the largest value of \(|2x + 5| \) for which this inequality holds true.

Interval of Convergence

The interval of convergence follows as a direct extension from the radius of convergence.It describes the actual values for which \( x \) makes the series converge.
To find this interval, solve the inequality derived during the root test:\[ |2x + 5| < 1 \]Breaking it into two inequalities:

• \( 2x + 5 < 1 \): Solve this to obtain \( 2x < -4 \), or \( x < -2 \).
• \( 2x + 5 > -1 \): Solve this to get \( 2x > -6 \), or \( x > -3 \).

By combining these, we find the interval of convergence as \( -3 < x < -2 \). This means the series converges for these values of \( x \). The interval does not include the endpoints, indicating convergence only occurs within this open range.

Absolute Convergence

Absolute convergence is a stronger form of convergence. A series is said to converge absolutely if the series of absolute values converges.
For the given series, absolute convergence occurs across the entire interval of convergence.

• Within the interval \( -3 < x < -2 \), the series absolutely converges. This is due to the results from applying the root test, producing a strict inequality that ensures no divergence.
• To verify, consider any \( x \) within \( -3 < x < -2 \); you'll findthat absolute values converge, supporting absolute convergence throughout this interval.

However, at endpoints \( x = -3 \) and \( x = -2 \), the series does not converge absolutely because the inequality \(|2x + 5| < 1 \) fails, as calculation shows \(|2x + 5| = 1\). Thus, absolute convergence is only within the specified interval.

Conditional Convergence

Conditional convergence is a more nuanced concept. It's where a series converges, but it does not converge absolutely.
For this series:

• The endpoints \( x = -3 \) and \( x = -2 \) need to be checked separately to see if the series converges conditionally.
• Given the results of our analysis:
  At these endpoints, the value \(|2x + 5| = 1 \) occurs, failing the root test's strict inequality, which implies no conditional convergence is possible here.

Therefore, the series does not converge conditionally at any of the endpoints. It only converges absolutely within the open interval \(-3 < x < -2\), leaving no room for conditional convergence.