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Old 12-16-2009, 04:58 PM   #1
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Default Testing for convergence/dvergence

I'm working with improper integrals. I have to use the direct or limit comparison test to see if the integral converges or diverges. Out book shows how with infinity. What about a asymptote at say, x=2? Does it work the same way?
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Old 12-16-2009, 05:16 PM   #2
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Default Re: Testing for convergence/dvergence

Not trying to be a dick but why would you post a question like that here? are you trying to show everyone how smart you are?

It would be more efficient to ask a teacher, fellow student or google. You should know that these forums are slow and that by the time you "Might" get a correct answer you could have found it on your own 100 times faster

Just saying
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Old 12-16-2009, 05:37 PM   #3
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Default Re: Testing for convergence/dvergence

Quote:
Originally Posted by ErdricK View Post
Not trying to be a dick but why would you post a question like that here? are you trying to show everyone how smart you are?
Because I know there are other people on here in much higher math classes that can help me

Quote:
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It would be more efficient to ask a teacher, fellow student or google. You should know that these forums are slow and that by the time you "Might" get a correct answer you could have found it on your own 100 times faster

Just saying
I tried google. but it its for series, and we haven't covered those yet. My teacher isn't available until tomorrow, and none of the other kids are on facebook and I don't have the number of anyone who would know.
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Old 12-16-2009, 06:55 PM   #4
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Default Re: Testing for convergence/dvergence

Quote:
Originally Posted by ErdricK View Post
Not trying to be a dick but why would you post a question like that here? are you trying to show everyone how smart you are?

It would be more efficient to ask a teacher, fellow student or google. You should know that these forums are slow and that by the time you "Might" get a correct answer you could have found it on your own 100 times faster

Just saying
Being in Calc 1 or 2 isn't supersmart... both are freshman/highschool level classes, gen ed.

I just finished my calc 1 final a couple minutes ago, but we didn't get very far into integrals... so I can't help you, sorry. Jogo will know it though, he is super good at math, hah
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Old 12-16-2009, 10:41 PM   #5
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Default Re: Testing for convergence/dvergence

Let me get this straight: You want to know whether a series converges/diverges as it approaches the value x=2?

Assuming that is what you want to know, I give you this answer: Nearly all series will converge when it approaches 2. Why? You have to recall that series generally start at 0 or 1, and they increase by integers. This does not give time for a series to be divergent (meaning the sum approaches infinity) unless the series itself contains infinity. The reason you use improper integrals to determine convergence/divergence is because you can write a power series as a function of x.

If you have an asymptote in the middle of the function that represents a power series, then what you have to do is separate the integral into two integrals. Then, you evaluate the first with the upper limit approaching the asymptotic value from the negative side, and you evaluate the second with the lower limit approaching the asymptotic value from the positive side. Once done, your answer should condense into infinity or a finite number.

I suggest that you be careful with the accuracy of my second paragraph. The conditions for the Integral Test are that f is a positive, continuous, and decreasing function (maybe has to be differentiable over the interval too). For this reason, you would not be able to apply it to a function that has an asymptote. But by what I know of math, if you split the interval into two intervals where f satisfies the above condition on both intervals, then you'd be able to apply the Integral Test, but this time, in order for f to diverge, you'd only need one of the two integrals to diverge. Does that make sense? I kind of rumbled because series is not my forte.
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Old 12-16-2009, 10:50 PM   #6
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Default Re: Testing for convergence/dvergence

Kinda. I'm talking about with the limits where if f(x)>g(x)>0 and f(x) converges then g(x) converges. And if limit of f(x)/g(x) as x approaches infinity is between 0 and infinity exclusive, then they both converge or diverge. My book explains both of those with using infinity only, and I'm wondering if it works for any other place where there is an improper integral.
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Old 12-16-2009, 10:54 PM   #7
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Default Re: Testing for convergence/dvergence

An integral is only improper when one of the limits is +- infinity (generally +). Take an integral from 0 to 2 where x = 2 is an asymptote. This is not an improper integral. This is simply an integral with an undefined answer. It seems to me that you're getting those two mixed up a bit.
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Old 12-16-2009, 11:07 PM   #8
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Default Re: Testing for convergence/dvergence

Our book says that if a function is discontinuous at any point in the integral it is improper.
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Old 12-16-2009, 11:44 PM   #9
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Default Re: Testing for convergence/dvergence

An improper integral is defined as an integral with limits ranging from c to infinity. What I think your book is trying to say is that the integral you're calculating is improper--not ideal to put it in other terms. It is so because there's an asymptote or some sort of discontinuity. To fix this, you break up the interval into 2 other intervals such that the new intervals construct 2 "proper" integrals.
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