Question

Dominance Relationships for Sequences: Order the following sequences by dominance when $a>0andb>1$

$\left\{k!\right\}$

Short answer

The required order of dominance sequence is$\ln k<<k^a<<b^k<<k!$

Step-by-step solution

Step 1. Given information

The given expression is$\left\{k!\right\}$

Step 2. Explanation

Use the concept of dominance relationships for simple sequences,

For any real number $a>0andb>1$, the following string of dominance holds.

$\ln k<<k^a<<b^k<<k!$ for k sufficiently large.