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

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

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'sSo 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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**boy15****Member**- Registered: 2010-03-29
- Posts: 16

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

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

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)*

Wrap it in bacon

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

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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