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Sequences and Series - Power Series Question

Hey, i have a tonne of these type of questions and i cannot for the life of me understand how they are done....

This is the answer to one of them which asks you to find the convergence and interval of convergence of the series: The sum from n=1 to infinity of the A(n) term at the start of the solution. I don't understand the step where they replace lim{[(n)/(n+1)]^3} with 1^3.

Not sure if i grasp the concept of what to do (Headbang)... can anyone explain?

Thanks.