Determine whether the infinite geometric series converges or diverges. If it converges, find its sum.
-$$\sum _ { \mathrm { k } = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }$$
A) Converges; $- 1.05$
B) Converges; $2.65$
C) Diverges
D) Converges; $1.05$

Question

Quizplus · Accepted Answer

The series converges because the common ratio $r = -0.9$ satisfies $|r| < 1$. The sum of an infinite geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Here, $a = -2$ and $r = -0.9$, so $S = \frac{-2}{1 - (-0.9)} = \frac{-2}{1 + 0.9} = \frac{-2}{1.9} = - \frac{2}{1.9} = - 1.05$.

Determine Whether the Infinite Geometric Series Converges or Diverges ∑k=1∞−2(−0.9)k−1\sum _ { \mathrm { k } = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }k=1∑∞​−2(−0.9)k−1

Determine Whether the Infinite Geometric Series Converges or Diverges $\sum _ { \mathrm { k } = 1 } ^ { \infty } - 2 ( - 0.9 ) ^ { \mathrm { k } - 1 }$