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 Post subject: Answers to the actual SOS Math stuffPosted: Tue, 15 Jul 2008 05:02:17 UTC
 S.O.S. Oldtimer

Joined: Sun, 8 Jun 2008 01:27:21 UTC
Posts: 173
Location: Pennsylvania

http://www.sosmath.com/calculus/series/ ... blems.html

Ok here is one of them for now

We see that this a decreasing series of all positive terms so we may apply the Integral Test

Well I do not know it exactly, but since

and for the last integral, let

So then

So

Using integration by parts on that integral we get

So now we see looking at

So

So we get

Now seeing as stated that

We have

Now seeing once again by

We can conclude that

So now back to the pertinent matter, using we can see that

This finally enables us to say that is convergent.

Last edited by Mathstud28 on Tue, 15 Jul 2008 07:51:45 UTC, edited 1 time in total.

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 Post subject: Posted: Tue, 15 Jul 2008 05:55:50 UTC
 S.O.S. Oldtimer

Joined: Sun, 8 Jun 2008 01:27:21 UTC
Posts: 173
Location: Pennsylvania
Here is another one

Question 14

Once again integral test

This last limit is undefined, therefore the integral is undefined, and moreover the series is divergent.

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 Post subject: Posted: Tue, 15 Jul 2008 07:34:26 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9632
Unfortunately the integral test is invalid for both series because the terms are not nonnegative.

For example, sin(4)/4 < 0 and cos(ln(6))/6 < 0.

Instead, you may want to consider Dirichlet's test.

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 Post subject: Posted: Tue, 15 Jul 2008 07:40:18 UTC
 S.O.S. Oldtimer

Joined: Sun, 8 Jun 2008 01:27:21 UTC
Posts: 173
Location: Pennsylvania
Matt wrote:
Unfortunately the integral test is invalid for both series because the terms are not nonnegative.

For example, sin(4)/4 < 0 and cos(ln(6))/6 < 0.

Instead, you may want to consider Dirichlet's test.

Hahahahahahahahahahahahahahahahahaha, and series are supposed to be my speciality . I cannot believe I missed that, well, at least people will see how to integrate .

Wow, that is depressing that I did all that work, it didn't even occur to me.

Thanks Matt.

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 Post subject: Posted: Tue, 15 Jul 2008 07:49:50 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Wed, 1 Oct 2003 04:45:43 UTC
Posts: 9632
Mathstud28 wrote:
Wow, that is depressing that I did all that work, it didn't even occur to me.

I know... I considered not replying because I did not want to spoil the fun. But it happens.

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 Post subject: Posted: Tue, 15 Jul 2008 07:52:36 UTC
 S.O.S. Oldtimer

Joined: Sun, 8 Jun 2008 01:27:21 UTC
Posts: 173
Location: Pennsylvania
Matt wrote:
Mathstud28 wrote:
Wow, that is depressing that I did all that work, it didn't even occur to me.

I know... I considered not replying because I did not want to spoil the fun. But it happens.

I am going to blame it on the lateness that I submitted it ::shifts eyes with insecurity::

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