Intervals of Convergence
In Exercises 1–36, (a) find the series’ radius and interval of convergence.
∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]

검증된 단계별 안내

Identify the general term of the series: \( a_n = \frac{(3x + 1)^{n+1}}{2n + 2} \).

To find the radius of convergence, apply the Root Test or Ratio Test. Here, the Ratio Test is convenient. Consider \( \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \).

Compute the ratio \( \left| \frac{a_{n+1}}{a_n} \right| = \left| \frac{(3x + 1)^{n+2}}{2(n+1) + 2} \cdot \frac{2n + 2}{(3x + 1)^{n+1}} \right| = \left| (3x + 1) \right| \cdot \frac{2n + 2}{2n + 4} \).

Take the limit as \( n \to \infty \) of the ratio: \( \lim_{n \to \infty} \left| (3x + 1) \right| \cdot \frac{2n + 2}{2n + 4} = \left| 3x + 1 \right| \cdot 1 = \left| 3x + 1 \right| \).

Set the limit less than 1 for convergence: \( \left| 3x + 1 \right| < 1 \). Solve this inequality to find the interval of convergence, then check the endpoints \( x \) values by substituting back into the original series to determine if the series converges at those points.

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주요 개념

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Radius of Convergence

The radius of convergence is the distance from the center of a power series within which the series converges absolutely. It can be found using the Ratio Test or Root Test on the general term of the series. This radius defines an interval on the x-axis where the series behaves well and sums to a finite value.

Interval of Convergence

The interval of convergence is the set of all x-values for which the power series converges. It is centered at the series’ expansion point and extends to the radius of convergence on both sides. Endpoints must be tested separately to determine if the series converges or diverges there.

Applying the Ratio Test to Power Series

The Ratio Test compares the limit of the absolute value of consecutive terms to determine convergence. For power series, it helps find the radius of convergence by simplifying the ratio of terms involving x. If the limit is less than one, the series converges; if greater, it diverges.

Intervals of ConvergenceIn Exercises 1–36, (a) find the series’ radius and interval of convergence.∑ (from n = 1 to ∞) [ (3x + 1)^(n + 1) / (2n + 2) ]