Determine if the series converges absolutely, converges, or diverges.
-$$\sum _ { n = 1 } ^ { \infty } ( - 5 ) ^ { - n }$$
A) converges absolutely
B) diverges
C) converges conditionally

Question

Quizplus · Accepted Answer

We use the geometric series test to determine the convergence of the given series. The series can be written as:
$$\sum _ { n = 1 } ^ { \infty } \left(\frac{-1}{5}\right)^n$$
This is a geometric series with first term $a=\frac{-1}{5}$ and common ratio $r=\frac{-1}{5}$. The formula for the sum of a geometric series is:
$$S = \frac{a}{1-r}$$

Using this formula, we get:

$$S = \frac{\frac{-1}{5}}{1-\frac{-1}{5}} = \frac{-1}{4}$$

Since the absolute value of the common ratio $\left|\frac{-1}{5}\right|$ is less than 1, the series converges absolutely. Therefore, the answer is A) converges absolutely.

Determine If the Series Converges Absolutely, Converges, or Diverges ∑n=1∞(−5)−n\sum _ { n = 1 } ^ { \infty } ( - 5 ) ^ { - n }n=1∑∞​(−5)−n

Determine If the Series Converges Absolutely, Converges, or Diverges $\sum _ { n = 1 } ^ { \infty } ( - 5 ) ^ { - n }$