Question

for the given geometric series, write out the first few terms of the series to find a and r, and find the sum of the series. then express the inequality |r| < 1 in terms of x and find the values of x for which the inequality holds and the series converges.

$$\sum_{n = 0}^{\infty} (-1)^n(x)^n$$

a = 1 and r = -x

find the sum.

$$\sum_{n = 0}^{\infty} (-1)^n(x)^n = \frac{1}{1 + x}$$

(type an expression using x as the variable.)

express the inequality |r| < 1 in terms of x.

| - x | < 1 (type an expression using x as the variable.)

the inequality holds and the series converges for □.

(simplify your answer. type a compound inequality.)

Explanation:

Step1: Simplify the absolute - value inequality

We know that \(|-x| = |x|\). So the inequality \(|-x|<1\) can be rewritten as \(|x|<1\).

Step2: Solve the absolute - value inequality

By the definition of absolute value, if \(|x|<1\), then \(-1 < x<1\).

Answer:

\(-1 < x<1\)