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

Explanation:

Step1: Substitute \( r=-x \) into \( |r|<1 \)

$$ |-x|<1 $$

Step2: Simplify the absolute - value inequality

Since \( |-x| = |x| \), the inequality becomes \( |x|<1 \)

Answer:

\( |x|<1 \)