Find the values of x for which the geometric series converges.
-$$\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$$
A) $\{ x \mid x$ is not a multiple of $2 \pi \}$
B) $\{ x \mid x$ is not a multiple of $\pi \}$
C) $- \infty

Question

Find the values of x for which the geometric series converges.
-$$\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$$
A) $\{ x \mid x$ is not a multiple of $2 \pi \}$
B) $\{ x \mid x$ is not a multiple of $\pi \}$
C) $- \infty < x < \infty$
D) diverges for all $x$

Quizplus · Accepted Answer

The series converges for all x because the common ratio, based on the absolute value of $1/(8 + \sin x)$, is always less than 1 regardless of x, ensuring convergence.

Find the Values of X for Which the Geometric Series ∑n=0∞(−1)n61(8+sin⁡x)n\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }n=0∑∞​6(−1)n​(8+sinx)n1​

Find the Values of X for Which the Geometric Series $\sum _ { \mathrm { n } = 0 } ^ { \infty } \frac { ( - 1 ) ^ { \mathrm { n } } } { 6 } \frac { 1 } { ( 8 + \sin \mathrm { x } ) ^ { \mathrm { n } } }$