In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.
∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ

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Recognize that the given series is a geometric series of the form \(\sum_{n=0}^\infty r^n\) where the common ratio \(r = \frac{(x^2 - 1)}{2}\).

Recall Theorem 20, which states that a geometric series \(\sum_{n=0}^\infty r^n\) converges if and only if \(|r| < 1\). Use this to find the interval of convergence by solving the inequality \(\left| \frac{(x^2 - 1)}{2} \right| < 1\).

Multiply both sides of the inequality by 2 to get \(|x^2 - 1| < 2\). This inequality will help determine the values of \(x\) for which the series converges.

Solve the inequality \(|x^2 - 1| < 2\) by considering the two cases: \(-2 < x^2 - 1 < 2\). Add 1 to all parts to get \(-1 < x^2 < 3\). Since \(x^2\) is always non-negative, the lower bound \(-1 < x^2\) is always true, so focus on \(x^2 < 3\) which implies \(-\sqrt{3} < x < \sqrt{3}\).

Within this interval, use the formula for the sum of a geometric series \(S = \frac{1}{1 - r}\), substituting \(r = \frac{(x^2 - 1)}{2}\) to express the sum as a function of \(x\).

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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 x-values for which a given power series converges. To find it, we often use the Ratio or Root Test to determine where the series converges absolutely. This interval may be open, closed, or half-open depending on endpoint behavior.

Geometric Series and Its Sum

A geometric series has the form ∑ arⁿ and converges when |r| < 1. Its sum is given by a/(1 - r). Recognizing a series as geometric allows us to find a closed-form expression for the sum within the interval of convergence.

Theorem 20 (Power Series Convergence and Sum)

Theorem 20 typically states conditions under which a power series converges and how to express its sum as a function within the interval of convergence. Applying this theorem helps identify the radius and interval of convergence and find the sum function explicitly.

In Exercises 43–50, use Theorem 20 to find the series’ interval of convergence and, within this interval, the sum of the series as a function of x.∑ (from n = 0 to ∞) [ (x² − 1) / 2 ]ⁿ