Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges, A) $u 7$ B) $6

Question

Find all positive values of $u$ for which the series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$ converges, A) $u > 7$ B) $6 < u < \frac { 7 } { e }$ C) $0 < u < \frac { 1 } { e }$ D) $u < 6$ E) $u > \ln 7$

Quizplus · Accepted Answer

The series is a geometric series with a common ratio $u$. For convergence, the absolute value of the common ratio must be less than 1, i.e., $0 < u < 1$. Additionally, the series involves a factor of $m$, which suggests a comparison to the exponential function, leading to the condition $0 < u < \frac{1}{e}$ for convergence.

Find All Positive Values Of uuu For Which the Series ∑m=1∞6um7m\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m∑m=1∞​6um7m

Find All Positive Values Of $u$ For Which the Series $\sum _ { m = 1 } ^ { \infty } 6 u ^ { m } 7 m$