Intervals of Convergence
In Exercises 1–36, for what values of x does the series converge (b) absolutely?
∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]

검증된 단계별 안내

Identify the given power series: \(\sum_{n=0}^{\infty} (-2)^n (n+1) (x-1)^n\).

To determine absolute convergence, consider the series formed by the absolute values of the terms: \(\sum_{n=0}^{\infty} |(-2)^n (n+1) (x-1)^n| = \sum_{n=0}^{\infty} (n+1) |2|^n |x-1|^n = \sum_{n=0}^{\infty} (n+1) (2|x-1|)^n\).

Recognize that this is a power series in terms of \(r = 2|x-1|\) with general term \((n+1) r^n\). To analyze convergence, use the root or ratio test, or recall the known result for the series \(\sum (n+1) r^n\).

Recall that the series \(\sum_{n=0}^{\infty} (n+1) r^n\) converges if and only if \(|r| < 1\). Therefore, set up the inequality \(2|x-1| < 1\) to find the interval where the series converges absolutely.

Solve the inequality \(2|x-1| < 1\) to find the values of \(x\) for which the original series converges absolutely. This will give the interval of absolute convergence.

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주요 개념

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Radius and Interval of Convergence

The radius of convergence defines the distance from the center point within which a power series converges. The interval of convergence includes all x-values for which the series converges, possibly including endpoints. Finding this interval involves applying tests like the Ratio or Root Test to the general term.

Absolute Convergence

A series converges absolutely if the series of absolute values converges. This is a stronger form of convergence ensuring the original series converges regardless of term signs. Testing absolute convergence often simplifies analysis by removing alternating signs.

Ratio Test for Series Convergence

The Ratio Test evaluates the limit of the absolute value of the ratio of consecutive terms. If this limit is less than one, the series converges absolutely; if greater than one, it diverges. This test is especially useful for power series involving factorials or exponential terms.

Intervals of ConvergenceIn Exercises 1–36, for what values of x does the series converge (b) absolutely?∑ (from n = 0 to ∞) [ (−2)ⁿ (n + 1) (x − 1)ⁿ ]