Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } | \sin 4 x | ^ { n }$$
A) $- \infty

Question

Find the values of x for which the geometric series converges.
-$$\sum _ { n = 0 } ^ { \infty } | \sin 4 x | ^ { n }$$
A) $- \infty < x < \infty$
B) $\{ x \mid x$ is not a multiple of $\pi \}$
C) $\{ x \mid x$ is not an odd multiple of $\pi / 8 \}$
D) diverges for all $x$

Quizplus · Accepted Answer

For a geometric series to converge, the common ratio must have an absolute value less than 1. In this case, the common ratio is $|\sin 4x|$, which will be less than 1 if and only if $\sin 4x$ is between $-1$ and $1$ (exclusive). This happens if and only if $x$ is not an odd multiple of $\frac{\pi}{8}$ (since at these values of $x$, $\sin 4x = \pm 1$). Therefore, the series converges for all $x$ except odd multiples of $\frac{\pi}{8}$. Thus, the correct answer is $\boxed{	extbf{(C)}}$.

Find the Values of X for Which the Geometric Series ∑n=0∞∣sin⁡4x∣n\sum _ { n = 0 } ^ { \infty } | \sin 4 x | ^ { n }n=0∑∞​∣sin4x∣n

Find the Values of X for Which the Geometric Series $\sum _ { n = 0 } ^ { \infty } | \sin 4 x | ^ { n }$