Find the indicated partial sum of the series.
$$\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$$
fourth partial sum
A) $\frac { 5 } { 81 }$

B) $\frac { 305 } { 81 }$

C) $- \frac { 100 } { 81 }$

D) $\frac { 35 } { 81 }$

E) $- \frac { 205 } { 162 }$

Question

Find the indicated partial sum of the series.
$$\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$$
fourth partial sum
A) $\frac { 5 } { 81 }$

B) $\frac { 305 } { 81 }$

C) $- \frac { 100 } { 81 }$

D) $\frac { 35 } { 81 }$

E) $- \frac { 205 } { 162 }$

Quizplus · Accepted Answer

The series is geometric with first term $a = -\frac{5}{3}$ and common ratio $r = -\frac{1}{3}$. The formula for the nth partial sum of a geometric series is $S_n = a \frac{1-r^n}{1-r}$. For the fourth partial sum, $S_4 = -\frac{5}{3} \frac{1-(-\frac{1}{3})^4}{1+\frac{1}{3}} = -\frac{100}{81}$.

Find the Indicated Partial Sum of the Series ∑i=1∞5(−13)i\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }i=1∑∞​5(−31​)i

Find the Indicated Partial Sum of the Series $\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$