Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$$3 - 24 + 192 - 1536 + \ldots + ( - 1 ) ^ { n - 1 } 3 \cdot 8 ^ { n - 1 } + \ldots$$
A) $\frac { 3 \left( 1 - ( - 8 ) ^ { n } \right) } { 1 - 8 } ; \frac { 3 } { 7 }$
B) $\frac { 3 \left( 1 - ( - 8 ) ^ { n } \right) } { 1 + 8 }$; series diverges
C) $\frac { 3 \left( 1 - ( - 8 ) ^ { n } \right) } { 8 - 1 } ;$ series diverges
D) $\frac { 3 \left( 1 - ( - 8 ) ^ { n } \right) } { 1 + 8 } ; \frac { 1 } { 3 }$

Question

Quizplus · Accepted Answer

The series is geometric with the first term 3 and common ratio -8. The formula for the nth partial sum of a geometric series is $ S_n = \frac{a(1 - r^n)}{1 - r} $. Here, $ a = 3 $ and $ r = -8 $, so the formula becomes $ \frac{3(1 - (-8)^n)}{1 + 8} $. Since the common ratio's absolute value is greater than 1, the series diverges.

Find a Formula for the Nth Partial Sum of the Series