Determine whether the geometric series converges or diverges. If it converges, find its sum.
$$\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$$
A) 30
B) 36
C) 5
D) Diverges

Question

Quizplus · Accepted Answer

The general term of the series is $a_n = 5^n6^{-n+1} = \frac{5^n}{6^n}\cdot 6 = (\frac{5}{6})^n\cdot6$. Since $|\frac{5}{6}|<1$, the series converges by the geometric series test. The sum is then
$$\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }=6\sum_{n=0}^\infty\left(\frac{5}{6}\right)^n=6\cdot\frac{1}{1-\frac{5}{6}}=36.$$Therefore, the answer is (B).

Determine Whether the Geometric Series Converges or Diverges ∑n=0∞5n6−n+1\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }n=0∑∞​5n6−n+1

Determine Whether the Geometric Series Converges or Diverges $\sum _ { n = 0 } ^ { \infty } 5 ^ { n } 6 ^ { - n + 1 }$