Evaluate the series, if it converges.
-$$\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }$$
A) $- 3$
B) Does not converge.
C) 3
D) 4

Question

Quizplus · Accepted Answer

This series is a geometric series with a common ratio $r = \frac{3}{4}$ and the first term $a = \left(\frac{3}{4}\right)^1 = \frac{3}{4}$. A geometric series converges if $|r| < 1$, which is true in this case. The sum of a converging geometric series is given by $S = \frac{a}{1 - r}$, where $a$ is the first term and $r$ is the common ratio. Substituting the given values, we get $S = \frac{\frac{3}{4}}{1 - \frac{3}{4}} = \frac{\frac{3}{4}}{\frac{1}{4}} = 3$.

Evaluate the Series, If It Converges ∑k=1∞(34)k\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }k=1∑∞​(43​)k

Evaluate the Series, If It Converges $\sum _ { \mathrm { k } = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { \mathrm { k } }$