Question

Determine if the improper integral converges or diverges. Justify your answer with a correct inequality and a conclusion.

Answer 1

\[\int\limits_{2}^{\infty}\frac{ x^{3}+1 }{ (x^{4}+4x+1)^{2} }dx\]

Answer 2

What's the degree of the denominator?

Answer 3

8? At least, the highest degree will be once it's FOILed out.

Answer 4

Right. What rational function can you compare it to, then?

Answer 5

...a quadratic? I'm sorry, I'm just a little confused over the 'rational function' part. 1/x^5?

Answer 6

Yeah, that's it. Rational is the same as fractional. So, by comparison with 1/x^5, you have
\[\frac{x^3+1}{x^8+\cdots+1}=\frac{1+\cdots}{x^5+\cdots} \le \frac{1}{x^5}\]

Answer 7

And then you can use the P-test. Awesome.
Is it generally appropriate to use the equals sign as you did above, or would it be better to use the approximately equal to sign instead?

Answer 8

Or do the ellipses account for extra things that would be necessary to make them equal?

Answer 9

The "or equal to" part is there because I didn't bother rewriting all the terms. I was only concerned with the leading term. It's probably not equal, so I think it's strictly "less than".

As for using "approximately," I don't think that would work because that would indicate the LHS is either greater than or less than the RHS. You need to show that it's strictly less than the RHS in order to make a conclusion about its convergence.

Answer 10

I meant more in terms of the two left terms, but I see what you mean.