I'm asked to determine if the following integral is convergent/divergent:

So I'm just wondering if I'm on the right track with what I'm doing:

As

So can I then just say this??:

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- Feb 3rd 2008, 10:53 AMTrevorPConvergent/Divergent Integrals
I'm asked to determine if the following integral is convergent/divergent:

So I'm just wondering if I'm on the right track with what I'm doing:

As

So can I then just say this??:

- Feb 3rd 2008, 11:55 AMmr fantastic
- Feb 3rd 2008, 12:00 PMKrizalid
It converges.

You may perform the straightforward integration by parts way, but I'll perform three ways to tackle this.

__Solution__#1:

Substitute

__Solution__#2:

Construct a double integral to reverse integration order, the rest follows easily.

__Solution__#3:

By the geometric series

This yields Hence

The last sum is a telescoping series. - Feb 3rd 2008, 03:28 PMmr fantastic
- Feb 3rd 2008, 03:35 PMTrevorP
Yeah the name l'Hopital seems familiar but I don't quite know what it is. I'll try a similar technique for the next few questions I have.

- Feb 3rd 2008, 04:07 PMJhevon
- Feb 3rd 2008, 04:58 PMTrevorP
Ok so here's my next one and my attempt on it:

So there must be a singularity at x = 0.

Also:

So:

Does this work??? - Feb 3rd 2008, 06:11 PMThePerfectHacker
- Feb 3rd 2008, 07:13 PMTrevorP
Ohh jeez. Thanks. So my logic makes sense then? Have I proved it's convergent?

- Feb 3rd 2008, 09:48 PMmr fantastic
- Feb 3rd 2008, 11:44 PMmr fantastic
Thanks for that useful thumbnail, Jhevon. I'll just add the following:

has the indeterminant form . The re-write to puts it into a 'standard' indeterminant form.

Note that ' '

(and you can count how many mathematical cows I've slaughtered here. But that's OK - slaughtering mathematical cows is something physicists do best ;) )