Don't understand this limit change in a ratio test

In summary, the conversation discusses the ratio test and how the limit is changed in step 1 to step 2. The discussion also covers the use of power series, limits, and reciprocals. The solution involves dividing both the numerator and denominator by n^2. The expert quickly recognizes this based on the highest degree expression in the numerator and denominator.
  • #1
saybrook1
101
4

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.
 

Attachments

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  • #2
They're not turning the n's into reciprocals. They're dividing both the numerator and denominator by ##n^2##
 
  • #3
Mastermind01 said:
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!
 
  • #4
saybrook1 said:
Okay, awesome. Thank you very much!

Glad to be of help. I think you should mark this as solved then.
 
  • #5
Mastermind01 said:
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.
 
  • #6
saybrook1 said:
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.
 
  • Like
Likes Mastermind01
  • #7
Mark44 said:
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!
 

FAQ: Don't understand this limit change in a ratio test

1. What is a ratio test and how does it relate to limits?

A ratio test is a mathematical tool used to determine the convergence or divergence of a series. It involves taking the limit of a ratio between consecutive terms in the series. If the limit is less than one, the series converges. If it is greater than one, the series diverges. The concept of limits is essential in the ratio test as it helps determine the behavior of the series as the number of terms increases.

2. What does it mean if the limit in a ratio test changes?

If the limit in a ratio test changes, it means that the behavior of the series is also changing. In other words, the series may converge or diverge depending on the value of the new limit. This change could be due to a change in the terms of the series or a change in the number of terms considered.

3. How do I know if I should use a ratio test to determine convergence?

A ratio test is most useful when dealing with series that involve powers or factorials. If the terms in the series involve these functions, then a ratio test is a good option to determine convergence. However, there are other tests, such as the integral test and the comparison test, that may also be appropriate depending on the type of series.

4. Can I use a ratio test to determine the exact value of a limit?

No, a ratio test can only determine the convergence or divergence of a series. It does not provide the exact value of the limit. However, if the series converges, it can be used to approximate the value of the limit by taking the sum of a finite number of terms in the series.

5. Are there any limitations to using a ratio test?

Yes, there are limitations to using a ratio test. It only works for series with positive terms, and the limit must exist for the test to be applicable. Also, in some cases, the ratio test may be inconclusive, and other tests must be used to determine convergence or divergence.

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