Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$7 + 14 + 28 + \ldots + 7 \cdot 2 ^ { n - 1 } + \ldots$
A) $\frac { 7 \left( 1 + 2 ^ { n } \right) } { 1 + 2 } ; \frac { 7 } { 3 }$

B) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 1 - 2 } ; 7$

C) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 2 - 1 } ;$ series diverges

D) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 1 - 2 }$; series diverges

Question

Find a formula for the nth partial sum of the series and use it to find the series' sum if the series converges.
-$7 + 14 + 28 + \ldots + 7 \cdot 2 ^ { n - 1 } + \ldots$
A) $\frac { 7 \left( 1 + 2 ^ { n } \right) } { 1 + 2 } ; \frac { 7 } { 3 }$

B) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 1 - 2 } ; 7$

C) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 2 - 1 } ;$ series diverges

D) $\frac { 7 \left( 1 - 2 ^ { n } \right) } { 1 - 2 }$; series diverges

Quizplus · Accepted Answer

This series is a geometric series with the first term $a = 7$ and common ratio $r = 2$. The formula for the nth partial sum of a geometric series is $S_n = a \frac{1 - r^n}{1 - r}$. Substituting $a = 7$ and $r = 2$ gives $S_n = 7 \frac{1 - 2^n}{1 - 2}$, which matches option D. Since $r > 1$, the series diverges, meaning it does not converge to a finite sum.

Find a Formula for the Nth Partial Sum of the Series