Test the series for convergence or divergence. Select the correct answer.
$$\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$$
A) The series is convergent.
B) The series is divergent.

Question

Quizplus · Accepted Answer

We can rewrite the series as:
$$\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }=\sum _ { m = 1 } ^ { \infty }\frac {(-1)^{m+1}\cdot 6^{m+1}} {2^{24m}}$$
Using the ratio test, we have:
$$\lim_{m 	o \infty} \left|\frac{\frac{(-1)^{m+2}\cdot 6^{m+2}} {2^{24(m+1)}}}{\frac{(-1)^{m+1}\cdot 6^{m+1}} {2^{24m}}}ight|=\lim_{m 	o \infty} \left|\frac{6}{2^{24}}ight|=\frac{6}{2^{24}}< 1$$
So the series converges by the ratio test.

Test the Series for Convergence or Divergence ∑m=1∞(−6)m+148m\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }m=1∑∞​48m(−6)m+1​

Test the Series for Convergence or Divergence $\sum _ { m = 1 } ^ { \infty } \frac { ( - 6 ) ^ { m + 1 } } { 4 ^ { 8 m } }$