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#1 2010-08-30 02:01:22

boy15
Member
Registered: 2010-03-29
Posts: 16

sequences help

Hi everyone, I need some help on some sequences questions. I tried them, but failed bad.

1. Prove that the function:

converges whenever the following holds: 


2.


If
for all
and
, then what are the
?

Any help on either of these question would be greatly appreciated. I tried 1, but I could not get the limit to evaluate to a finite number, I must be doing it wrong.

Last edited by boy15 (2010-08-30 23:11:00)

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#2 2010-08-30 06:28:24

TheDude
Member
Registered: 2007-10-23
Posts: 361

Re: sequences help

I believe #1 can be proven with the ratio test.  I'm probably going to butcher the terminology but this should be understandable.

The function A is equivalent to the series

Define a new function S

So A(x) = S(x).  Now the ratio test says that if

then S(x) will converge.  So what is

?  Well, it's

So we know that S(x) converges, which of course means A(x) does as well since they're equal.

I read #2 differently.  I think it's trying to say

That would make it far, far more difficult than #1 though, so I'm not sure.  In any case it's well beyond me so I'm afraid I can't help on that one.


Wrap it in bacon

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#3 2010-08-30 23:09:34

boy15
Member
Registered: 2010-03-29
Posts: 16

Re: sequences help

TheDude wrote:

Thanks for the help, but I don't really understand how

.

Yeah, I think it question 2 should be interpreted your way but I still can't do anything with it. Can someone help with q2 please?

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#4 2010-08-30 23:45:16

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: sequences help

The second question is a lot simpler than you think.

From the first question, we know that A(x) converges whenever |x| < 1/2.
We can work out that in this range, A(x) = 1/(1-2x).

From there it's obvious what B(x) needs to be.


Why did the vector cross the road?
It wanted to be normal.

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#5 2010-08-31 02:24:37

TheDude
Member
Registered: 2007-10-23
Posts: 361

Re: sequences help

boy15 wrote:

Thanks for the help, but I don't really understand how

.

This is given to us.  The question is to prove that A(x) converges whenever

and since

we know that S(x) converges, which means A(x) also converges since they're equal.

mathsy, regarding question 2 that makes sense for |x| < 1/2, but I don't see a restriction in the question that x must be chosen such that A(x) converges.  My understanding (based on a quick look on Wikipedia) is that a Cauchy product can converge even if one of series being multiplied diverges.  Although now that I've taken a second look I can see that this is never stated anywhere in the article, so perhaps both series do have to converge for the Cauchy product to converge?

Last edited by TheDude (2010-08-31 02:25:21)


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#6 2010-08-31 14:52:11

boy15
Member
Registered: 2010-03-29
Posts: 16

Re: sequences help

mathsyperson: Thanks heaps, I was over complicating q2. I understand it now.

TheDude: Yeah, I missed that, I normally see it a_n/a_{n+1), but I understand it all now.

Thank you TheDude and mathsyperson for helping me.

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