Perhaps your usage of L' Hopital was inappropriate.
Pardon the notation, but...
Why do you need any fancy theorem?
I could use some help with an integral. The problem asks to determine whether a given integral converges or diverges. I know that the answer is "diverges" but I'm not sure why.
I used integration by parts to get:
Then I evaluated
I understood to require the use of L' Hopital's rule which I applied as follows: If this were true, the improper integral would evaluate to -1 - 0 = -1 and would converge. Can anyone tell me where I went wrong?
Well,
Is not true.
But, just noticing and going back to your integration by parts, the above is not true as well (you did the by parts incorrectly).
.........I did the limits on this before on your by parts, but since TKHunny already did this and I don't feel like doing it again with the correct equation I will leave it be. His answer satisfies why.
I realized after I posted that, my careless integration errors, i.e., missing the factor of 1/2.
But my real error was interpreting 1/(e^-2b) as b approaches negative infinity as resulting in 1/e^infinity rather than e^infinity. Thank you for clearing things up all, I'm okay now, I think!!