Don't understand this limit change in a ratio test

[saybrook1]

Homework Statement

I would like to understand how the limit was changed in the ratio test from step 1 to step 2 in the image that I've posted. I thought that the denominator would look like (2/n+2)(2/n+1) in step 2 since it looks like we are just turning the n's into reciprocals. Any help here would be greatly appreciated. Thank you very much as always.

Homework Equations

Power series, limits, reciprocals.

The Attempt at a Solution

I've tried to figure out why the argument in the second set of parentheses in the denominator becomes (2+1/n) instead of (2/n+1) once the limit is changed from infinity to zero in between steps 1 and 2. Just with brute force algebra I couldn't figure out how they went from step 1 to 2.

 

[Mastermind01]

They're not turning the n's into reciprocals. They're dividing both the numerator and denominator by $n^2$

 

[saybrook1]

They're not turning the n's into reciprocals. They're dividing both the numerator and denominator by $n^2$

Okay, awesome. Thank you very much!

 

[Mastermind01]

Okay, awesome. Thank you very much!

Glad to be of help. I think you should mark this as solved then.

 

[saybrook1]

Glad to be of help. I think you should mark this as solved then.

I marked this as solved, but can I ask you how you knew that they were dividing the numerator and denominator by n^2 right away? I think I would have to expand everything to see that. Thanks again.

 

[Mark44]

I marked this as solved, but can I ask you how you knew that they were dividing the numerator and denominator by n^2 right away? I think I would have to expand everything to see that. Thanks again.

Because the highest degree expression (in n) in the numerator is n² plus lower-degree terms, and the denominator is also 2nd-degree in n.

 

[saybrook1]

Because the highest degree expression (in n) in the numerator is n² plus lower-degree terms, and the denominator is also 2nd-degree in n.

Okay, great. Thanks!