Convergence Question?

Question

The question was does the integral of dx/(x^2+3x+2) from 0 to infinity converge or diverge, and if it converges what does it converge to.
I did the integral and all the work and came to:
limit as b approaches infinity of [ln(b+1)-ln(b+2)]-[ln1-ln2].
The correct answer is it converges to ln2. I can see how that answer is produced, the ln of infinity must come to a small number, and ln1 is zero leaving just a positive ln2. But how do I know that the ln(infinity) converges instead of diverges?

Answer 1

evaluate the limit as b approaches infinity

lim ln(b+1) - ln(b+2) ... this is an indefinite limit.. infinity minus infinity

= lim ln [(b+1)/(b+2)]
= ln lim [(b+1)/(b+2)]
= ln 1
= 0

... ok... i answered it for you... §

Answer 2

To do that, you have to recall that

[ln(b+1)-ln(b+2)] = ln [(b+1)/(b+2)]

The limit of the simplified expression above, when b approaches infinity is ln 1 or, (you're definitely right) 0. Thus it really ends up as positive ln 2. To actually determine the limit (as direct substitution leads to an indeterminate form), you could use L' Hopital's Rule or basic limit determination techniques.

Hope that helps ;-)

Answer 3

Log(b+1) - Log(b+2) is the same as

Log((b+1)/(b+2))

which means as b approaches infinity, we have Log(1), which is 0, and you're left with Log(2)

Answer 4

Because of the variance in destabilation dummy.