Intervals of Convergence
Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

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Identify the given series: \( \sum_{n=1}^{\infty} \frac{(3x + 1)^{n+1}}{2n + 2} \). We want to find the values of \( x \) for which this series converges absolutely.

To check for absolute convergence, consider the series of absolute values: \( \sum_{n=1}^{\infty} \left| \frac{(3x + 1)^{n+1}}{2n + 2} \right| = \sum_{n=1}^{\infty} \frac{|3x + 1|^{n+1}}{2n + 2} \).

Since the denominator \( 2n + 2 \) grows linearly and does not affect the radius of convergence, focus on the term \( |3x + 1|^{n+1} \). Use the root test or ratio test to find the radius of convergence with respect to \( |3x + 1| \).

Apply the root test: compute \( \lim_{n \to \infty} \sqrt[n]{\frac{|3x + 1|^{n+1}}{2n + 2}} = \lim_{n \to \infty} \frac{|3x + 1|^{(n+1)/n}}{(2n + 2)^{1/n}} = |3x + 1| \), since the other terms tend to 1.

Set the root test limit less than 1 for absolute convergence: \( |3x + 1| < 1 \). Solve this inequality for \( x \) to find the interval of absolute convergence.

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주요 개념

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Interval of Convergence

The interval of convergence is the set of all x-values for which a given power series converges. It is found by determining where the series converges absolutely or conditionally, often using tests like the Ratio or Root Test. This interval can be finite or infinite and includes checking endpoints separately.

Absolute Convergence

A series converges absolutely if the series of absolute values converges. Absolute convergence implies convergence regardless of the sign of terms, making it a stronger form of convergence. For power series, testing absolute convergence helps identify the radius and interval of convergence.

Ratio Test

The Ratio Test determines convergence by examining the limit of the ratio of consecutive terms' absolute values. If this limit is less than 1, the series converges absolutely; if greater than 1, it diverges. It is especially useful for power series to find the radius of convergence.

Intervals of ConvergenceIntervals of ConvergenceIn Exercises 1–36, for what values of x does the series converge (b) absolutely?∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]