Limit Comparison/Comparison Test on Non-rational functions

In summary, the series of (root(n^4 +1) - n^2) n goes from 1 to infinity. The Attempt at a Solution shows that you should multiply the expression by (sqrt(x^4+1)+n^2)/(sqrt(x^4+1)+n^2) and do some algebra in the numerator. Then see what you think.
  • #1
chrischoi614
7
0

Homework Statement


Either the Comparison Test or Limit Comparison Test can be used to determine whether the following series converge or diverge.
which test you would use (CT or LCT)
[ii] which series you would use in the comparison.
[iii] does the series converge or not

The series of (root(n^4 +1) - n^2) n goes from 1 to infinity

2. Relevent equations
Series of 1/n^2? I am not too sure

The Attempt at a Solution



So what i did was drag out the n^2 from the root so it becomes (n^2)(root(1+(1/n^4)))
and I know i Think i have to compare this with 1/n^2 , I know this series converge, but however I do not know how to explain correctly, to compare it with 1/n^2, if 1/n^2 really is the right one to compare to, or should i be using limit comparison test? I am quite lost at the moment, I have tried everything, but the fact that all I can use is CT and LCT, I really don't know how to solve it. I know that root (n^4 + 1) is just really close to n^2, its that (+1) that make this series happen... Pleasee and thanks :)
 
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  • #2
I would start by multiplying your expression by (sqrt(x^4+1)+n^2)/(sqrt(x^4+1)+n^2) and doing some algebra in the numerator. Then see what you think.
 
  • #3
I actually did that before, but i ended up with 2n^4 -(2n^2)(root(n^4 +1)) + 1 in the numerator, the fact that the square root is there really is making me struggle cus i don't know how to simplify it =\
 
  • #4
chrischoi614 said:
I actually did that before, but i ended up with 2n^4 -(2n^2)(root(n^4 +1)) + 1 in the numerator, the fact that the square root is there really is making me struggle cus i don't know how to simplify it =\

Then show us the algebra you did to get that. It's not right.
 
  • #5
no... i didnt get it wrong :S... i just put the terms together...
 
  • #6
chrischoi614 said:
no... i didnt get it wrong :S... i just put the terms together...

I'm glad you are so confident but (sqrt(x^4+1)-n^2)*(sqrt(x^4+1)+n^2) doesn't have a sqrt in it if you expand it. Show us how you got 2n^4-(2n^2)(root(n^4 +1))+1 or we can't help you. Did you not change the sign on the n^2? That's the whole 'conjugate' thing that makes the sqrt cancel.
 
Last edited:

1. What is the purpose of using the Limit Comparison/Comparison Test on Non-rational functions?

The Limit Comparison/Comparison Test is used to determine whether a given non-rational function converges or diverges by comparing it to a known function whose convergence or divergence is already known. This test is particularly useful when the non-rational function is complex and difficult to analyze directly.

2. How does the Limit Comparison/Comparison Test work?

The test involves taking the limit as x approaches infinity of the ratio of the given function and the known function. If this limit is a finite, non-zero number, then the two functions have the same convergence or divergence behavior. If the limit is zero or infinity, then the two functions have different convergence or divergence behavior.

3. What is the difference between the Limit Comparison Test and the Comparison Test?

Although both tests involve comparing a given function to a known function, the main difference is in the type of functions that can be compared. The Comparison Test is used for series of non-negative terms, while the Limit Comparison Test can be used for non-rational functions that may not necessarily be non-negative.

4. Can the Limit Comparison/Comparison Test be used for all non-rational functions?

No, the test may not be applicable to all non-rational functions. It is important to check whether the given function satisfies the conditions for the test, such as being positive and decreasing for all values of x greater than some number.

5. What are some common examples of non-rational functions that can be analyzed using the Limit Comparison/Comparison Test?

Some common examples include exponential functions, logarithmic functions, trigonometric functions, and polynomial functions. These types of functions often appear in series and integrals, making the Limit Comparison/Comparison Test a useful tool for analyzing their convergence or divergence behavior.

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