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## #1 2006-03-05 16:51:50

Alk
Member
Registered: 2006-03-05
Posts: 2

### Comparison Theorem

The problem I am working on is:

Use the comparison theorem to determine whether the integral is convergent of divergent. The integral is dx/(x + e^2x) from 1 to infinite.

My question is why does this not work.
For x >= 1, 1/(x+ e^2x)   <    1/x, therefore 1/x+e^2x is divergent.

For x >= 1, 1/(x+e^2x)   <   1/e^2x, therefore 1/x+e^2x is convergent.

Why doesnt mine work?

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## #2 2006-03-05 17:47:54

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Comparison Theorem

The comparison test is as follows:

If f(x) < g(x) and g(x) is convergent, then f(x) must be convergent

If f(x) > g(x) and g(x) is divergent, then f(x) must be divergent.

So you can't use < for divergent like you did.

Last edited by Ricky (2006-03-05 17:48:27)

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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## #3 2006-03-05 18:23:45

Alk
Member
Registered: 2006-03-05
Posts: 2

### Re: Comparison Theorem

I don't think I understand the actual process of determining this solution. So far I have
f(x) >= g(x) >= 0

f(x) = dx/(x + e^2x)

Would someone be willing to explain the steps in solving this?

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## #4 2006-03-05 18:29:54

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

### Re: Comparison Theorem

And g(x) is 1/x

So f(x) <= g(x)

But g(x) is divergent.  Thus, but the comparison test, this says absolutely nothing about convergence.

Basically, the covergent thm just says that if you have something less than a finite number (i.e. convergent), it is also a finite number.  If you have something greater than an infinite number (divergent), it is also infinite.

But what if you have something greater than a finite number?  You can't really tell if that's finite or infinite.  And if you have something less than an infinite number?  You can't really say if that is still finite or infinite.

"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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