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

검증된 단계별 안내

Rewrite the series to identify its general term clearly: \( a_n = \frac{(3x + 1)^{n+1}}{2n + 2} \). This helps in analyzing the convergence behavior.

Determine the radius and interval of convergence by applying the Ratio Test. Compute \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \) and simplify the expression to find the values of \( x \) for which the limit is less than 1.

Find the interval of convergence from the inequality obtained in the Ratio Test. This gives the open interval where the series converges absolutely.

Check the endpoints of the interval separately by substituting them back into the original series. Since the question asks for conditional convergence, test whether the series converges at these endpoints but not absolutely.

Use appropriate convergence tests (such as the Alternating Series Test or the p-series test) on the series at the endpoints to determine if the convergence is conditional there.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Interval of Convergence

The interval of convergence is the set of all x-values for which a given power series converges. To find it, one typically uses the Ratio or Root Test to determine the radius of convergence, then checks the endpoints separately to see if the series converges there.

Conditional vs. Absolute Convergence

A series converges absolutely if the series of absolute values converges; otherwise, if the series converges but not absolutely, it converges conditionally. Conditional convergence often occurs at the endpoints of the interval of convergence and requires tests like the Alternating Series Test.

Convergence Tests for Series

Tests such as the Ratio Test, Root Test, and Alternating Series Test help determine whether a series converges or diverges. The Ratio Test is useful for power series, while the Alternating Series Test can confirm conditional convergence when terms alternate in sign and decrease in magnitude.

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