For which positive integers $k$ is the series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$ convergent?
A) $k \geq 5$
B) $k \leq 0$
C) $k \geq 0$
D) $k \geq 1$
E) $k \leq - 5$

Question

Quizplus · Accepted Answer

The ratio test shows that the series converges for $k \geq 5$ because the factorial in the denominator grows faster than the factorial in the numerator when $k$ is large enough.

For Which Positive Integers kkk Is the Series ∑n=1∞(n!)5(kn)!\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }∑n=1∞​(kn)!(n!)5​

For Which Positive Integers $k$ Is the Series $\sum _ { n = 1 } ^ { \infty } \frac { ( n ! ) ^ { 5 } } { ( k n ) ! }$