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

Question

Quizplus · Accepted Answer

The series is a geometric series with the first term $a = -2$ and common ratio $r = -0.9$. A geometric series converges if the absolute value of the common ratio is less than 1, which is true in this case ($|-0.9| < 1$). The sum of a converging geometric series is given by $S = \frac{a}{1 - r}$, so substituting $a = -2$ and $r = -0.9$ gives $S = \frac{-2}{1 - (-0.9)} = \frac{-2}{1 + 0.9} = \frac{-2}{1.9} = -\frac{2}{1.9} = -1.05$.

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