Use when inequality is obvious
Imagine two people depositing money.
Person A deposits:
$100+$50+$25+⋯ and ends up with a finite amount.
Person B always deposits less than Person A.
Can Person B somehow accumulate an infinite amount? No.
If the bigger total stays finite, the smaller one must stay finite too.
Comparing growth rates. Use when terms are messy rational functions
If one series converges, multiplying by 2 won't suddenly make it diverge.
If one diverges, dividing by 2 won't suddenly make it converge.
So they must behave the same.