Question

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

Accepted Answer

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 $$|r| \u003c 1. $$Use this to find the interval of convergence by solving the inequality $$\left| \frac{(x^2$ - 1)}{2} \right| \u003c 1. $$Multiply both sides of the inequality by 2 to get $$|x^2$ - 1| \u003c 2. $$This inequality will help determine the values of $$x$$ for which the series converges. Solve the inequality $$|x^2$ - 1| \u003c 2 by $$considering the two cases: $$-2 \u003c x^2$ - 1 \u003c 2. $$Add 1 to all parts to get $$-1 \u003c x^2$ \u003c 3. $$Since $$x^2 is $$always non-negative, the lower bound $$-1 \u003c x^2$ is always true, so focus on x^2$ \u003c 3$$ which implies $$-\sqrt{3} \u003c x \u003c \sqrt{3}. $$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 $$x.$$

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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Key Concepts

Interval of Convergence

Geometric Series and Its Sum

Theorem 20 (Power Series Convergence and Sum)