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Don't understand this limit change in a ratio test

  1. Jan 3, 2017 #1
    1. The problem statement, all variables and given/known data
    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 in to reciprocals. Any help here would be greatly appreciated. Thank you very much as always.

    2. Relevant equations
    Power series, limits, reciprocals.

    3. 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.
     

    Attached Files:

  2. jcsd
  3. Jan 3, 2017 #2
    They're not turning the n's into reciprocals. They're dividing both the numerator and denominator by ##n^2##
     
  4. Jan 3, 2017 #3
    Okay, awesome. Thank you very much!
     
  5. Jan 3, 2017 #4
    Glad to be of help. I think you should mark this as solved then.
     
  6. Jan 3, 2017 #5
    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.
     
  7. Jan 3, 2017 #6

    Mark44

    Staff: Mentor

    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.
     
  8. Jan 3, 2017 #7
    Okay, great. Thanks!
     
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