Find all positive values of u for which the series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$ converges. A) $4

Question

Find all positive values of u for which the series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$ converges. A) $4 < u < \frac { 7 } { e }$ B) $0 < u < \frac { 1 } { e }$ C) $u > \ln 7$ D) $u > 7$ E) $u < 4$

Quizplus · Accepted Answer

The series converges if the term $3 u^{\ln 9 m}$ tends to zero as $m$ approaches infinity. This happens when the base $u^{\ln 9}$ is less than 1, which implies $u < \frac{1}{e}$.

Find All Positive Values of U for Which the Series ∑m=1∞3uln⁡9m\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }∑m=1∞​3uln9m

Find All Positive Values of U for Which the Series $\sum _ { m = 1 } ^ { \infty } 3 u ^ { \ln 9 m }$