Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

Limit of a recursive sequence

I'm stuck here and was hoping someone could give me a push in the right direction.

I'm playing with a sequence:

and I'm looking for

Any advice/answers would be very much appreciated.

Thanks

Last edited by bossk171 (2009-11-12 10:37:31)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Limit of a recursive sequence

Well, here's a few terms:

k_1 = 0.5
k_10 = 0.1707836...
k_100 = 0.1206115...
k_1000 = 0.1155092...
k_10000 = 0.1149988...
k_100000 = 0.1149477...
k_1000000 = 0.1149426...

So it seems like the sequence converges to some non-zero limit. No idea how to get it analytically though.

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Limit of a recursive sequence

Hi bossk171;

That recurrence is not a homework assignment.  That is a non linear difference equation with a trig term.Problems like that rarely have nice solutions. There is no analytical solution that I know off. That makes coming up with the limit you require difficult or impossible. Best I can do is provide a numerical answer.

The recurrence is equivalent to this infinite product:

I ran my own numerics using an idea like mathsyperson's only to more places. Then I took that and used shanks accelerators to speed up the convergence, I got an answer of.

We can assume that you have at least 20 digits of accuracy. I then tried to PSLQ this answer and get it in terms of known constants. This did not succeed. If I get anything else I will post. Hope this helps a little

Last edited by bobbym (2009-11-13 13:40:49)

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ken tutor

Guest

Re: Limit of a recursive sequence

http://math.berkeley.edu/~jtener/pdf/1bfa08/recursive%20sequence%20example.pdf

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Limit of a recursive sequence

Hi ken;

Welcome to the forum!

I was aware of that, the trouble is I could not get that method to work. You have an idea?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Limit of a recursive sequence

0
in most of times the cosine is smaller than 1, and in the rest of times it is 1. So the absolute value converges to 0 and so do the sequence itself.

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X'(y-Xβ)=0