Math Is Fun Forum

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Recursive sequence limit

Hi!
I have been stuck with this sequence:

I need to find an expression for the values of a and b for which the sequence converges to 0.

Any suggestion or solution would be very welcome!

Thank you!

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Hi gftrigo,

Solving the recurrence gives:

So we need to solve the inequalities:

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gAr;

The second latex statement is not coming through, can you fix it.

---

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Hi bobbym,

I'm also checking, I cannot spot the mistake.
In codecogs, it's fine!

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gAr;

I took care of it. Sometimes you cannot split lines up in one statement.

Hi gftrigo;

One such ordered pair of (a,b) is a = -1 and b = - 1/4 which will oscillate and converge to 0.

Solving the recurrence as gAr has already done for you. Search for pleasant a and b's in these intervals

---

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Hi bobbym,

Thanks for fixing it and solving the inequalities.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gAr;

One question what did you think about to get -1 and 1 as the intervals for those inequalities?

---

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.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Hi bobbym,

x^k ->0 as k->∞ when -1<x<1 . So I wrote that.
But I haven't dealt with simultaneous inequalities. Did I get the inequalities right?

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gAr;

It sure looks like it.

---

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.

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Re: Recursive sequence limit

Hi! Wow! thank you!

gAr can you explain me how you solved the recurrence to find that expression for ak (on post #2)?

Thanks!

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Hi gftrigo,

You're welcome.
I used a CAS. If you are interested to use that, you may download maxima: http://maxima.sourceforge.net/

Enter this into maxima:
load(solve_rec);
solve_rec(x[k]=%a*x[k-1]+%b*x[k-2],x[k]);

I have forgotten to solve it by hand, I'll try to remember.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

Ok, got it.

Write the characteristic equation:

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Re: Recursive sequence limit

Thank you gAr!

yes, the roots of the characteristic polynomial, I remember now!

I was trying to solve the inequalities of abs(r1)<1 and abs(r2)<1 but all I can get is:

with abs(r1)<1 I get:

which also gives

from abs(r2)<1 I get:

which gives

Is this right? I can't seem to make the final step to get what bobbym got.

gAr

Member

Registered: 2011-01-09

Posts: 3,482

Re: Recursive sequence limit

You're welcome gftrigo.

Bobbym can answer that better, I'm learning about inequalities.

---

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense"  - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Re: Recursive sequence limit

Thank you a lot gAr!

bobbym can you explain me the inequalities please?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gftrigo;

I need to find an expression for the values of a and b for which the sequence converges to 0.

I used a CAS also to get that answer. Your original question was to find an a and b that converged to 0. When working on a problem like that I do not usually stop my train of thought by working on sub problems that are not really connected to the problem.

---

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.

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Re: Recursive sequence limit

Ok. Thank you!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gftrigo;

Your welcome and your reply shows me that it is not a homework assignment. So if you want to know how to do that particular part of the answer for your own knowledge then I would gladly work on it and try see what Mathematica the brilliant did.

---

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.

gftrigo

Member

Registered: 2011-03-17

Posts: 7

Re: Recursive sequence limit

Hi bobbym!

Indeed it is not a homework, I really need to understand the steps that lead to the answer.
I really appreciate your help and patience!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Recursive sequence limit

Hi gftrigo;

It is easy enough to get the b<(1-a) and b<(1+a) but I am unable to solve them simultaneously and get his full answer.

---

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.