Integral convergence and limit comparison test

In summary, the conversation discusses using the limit comparison test for integrals in cases where the denominator is f(x) - c, and how it can be used to determine convergence. The conversation also mentions the relationship between the integral test and the limit comparison test for series and how they can be used interchangeably in some cases. It is also noted that the integral test can be used to determine convergence for series as well.
  • #1
Benny
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0
Hi, I've been thinking about the comparison test for integrals. Usually when I have an integrand where the denominator is f(x) + something positive I can usually find a suitable bound without much trouble. However when the denomoninator of the integrand is f(x) - something positive finding a bound is generally more difficult. This is because I would generally have no idea as to whether the integral converges or not.

So one day I was thinking about the limit comparison test which comes in handy when I have series' where the denominator is something like 2^n - 1. Long story short I used the limit comparison test with integrals by considering the integral as a series(in the mathematically incorrect way) and for a few examples it seemed to work. For example if I have a integral where the integrand is of the form 1/(f(x) - c), c > 0 I just wrote the series(I've omitted the sigma etc) 1/(f(n) - c) and used the limit comparison test for series to determine whether or not the series 1/(f(n)-1) converges. In the few examples I tried, if the corresponding series converges I've also found that the integral converges.

Can the limit comparison test for series be used as a test for integrals in this way? I know that there is an integral test for series but I'm not sure if there is anything the other way around.
 
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  • #2
I think you're a bit confused here...

The integral test for a series determines if a series converges/diverges. The limit comparison test does the same thing to a series. At times, it is preferred to use one over the other if the series is complicated or simple.

So if the question you're asking is whether they can "replace" one another, then I guess the answer is yes.
 
  • #3
TimNguyen said:
I think you're a bit confused here...

The integral test for a series determines if a series converges/diverges. The limit comparison test does the same thing to a series. At times, it is preferred to use one over the other if the series is complicated or simple.

So if the question you're asking is whether they can "replace" one another, then I guess the answer is yes.
I think what is meant is does the integral test for series work the other way. The answer is yes, the theorem can be formulated as the integral and the series converge or diverge together. Then oncce the question of does this integral coverge is exchanged for does this sum converge any method can be used, including comparison.
http://mathworld.wolfram.com/IntegralTest.html
http://mathworld.wolfram.com/ConvergenceTests.html
http://mathworld.wolfram.com/LimitComparisonTest.html
 
  • #4
Thanks for clearing that up.
 

FAQ: Integral convergence and limit comparison test

What is the Integral Convergence Test?

The Integral Convergence Test is a method used to determine if an infinite series converges or diverges by comparing it to a known convergent or divergent integral. It is based on the idea that if a function is decreasing and bounded, its improper integral will converge.

How do you use the Integral Convergence Test?

To use the Integral Convergence Test, you first need to identify a function that is greater than or equal to the terms of the series you are testing. Then, you need to integrate this function and evaluate the resulting integral. If the integral converges, then the series also converges. If the integral diverges, then the series also diverges.

What is the Limit Comparison Test?

The Limit Comparison Test is a method used to determine the convergence or divergence of an infinite series by comparing it to a known series. It is based on the idea that if the limit of the ratio of two series is a positive finite number, then the two series will have the same behavior.

How do you use the Limit Comparison Test?

To use the Limit Comparison Test, you first need to identify a known series that has a similar behavior to the series you are testing. Then, you need to take the limit of the ratio of the two series as n approaches infinity. If the limit is a positive finite number, then the two series behave in the same way and will either both converge or both diverge.

When should I use the Integral Convergence Test versus the Limit Comparison Test?

The Integral Convergence Test is typically used when the series being tested consists of positive terms and can be easily integrated. The Limit Comparison Test is used when the series being tested does not have easily recognizable terms, making it difficult to find a known series to compare it to. It is also useful when the series has alternating signs or when the terms have complicated expressions involving factorials or exponents.

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