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

question 33

Multiple Choice

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) \(k \leq - 5\)

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Definitions:

Absolute Value Inequality

An inequality equation involving the absolute value of a variable or expression, indicating a range of possible values.

Verbal Statement

A mathematical statement expressed in words.

Absolute Value Equation

An equation that contains an absolute value expression, representing the distance of a number from zero on the real number line.

Set Notation

A standardized method of describing or defining a set by clearly specifying its elements or the properties that its members must satisfy.