Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }$ converges. Select the correct answer. A) $u 7$ B) $6

Question

Find all positive values of $u$ for which the series $\sum _ { m - 1 } ^ { \infty } 6 u ^ { \mathrm { m } 7 m }$ converges. Select the correct answer. 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 common ratio $u^{7m}$. For convergence, the absolute value of the common ratio must be less than 1, i.e., $u^7 < 1$, which implies $0 < u < \frac{1}{e}$.

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

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