Math Is Fun Forum

jhomme

Member

Registered: 2011-06-02

Posts: 3

Limit of Quotient of Infinite Series

I’m trying to find an expression for the limit below.  I know a limit exists because I can calculate it in Excel but I’d like to find a closed-form version.  Any ideas on how to evaluate this?

a: = sum of (1+g)^(i-1) * (1-p)^(n-i), where i=1..n
b:= sum of (1+g)^(i-1), where i=1..n

evaluate limit as n-->infinity of {a/b}

Assume that p and g are constants such that:
0<p<1
-1<g<1

The quotient is essentially a weighted average where the weight factor is (1-p)^(n-i).  Larger values of i will have greater weight in the average.

Thanks!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Limit of Quotient of Infinite Series

Hi jhomme;

Is this what we are working on?

Where you want to evaluate the limit.

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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.

jhomme

Member

Registered: 2011-06-02

Posts: 3

Re: Limit of Quotient of Infinite Series

Yes, indeed!  Well done.  This is already a huge simplification that can help me quickly approximate the limit using high values of n.  That's 99% of what I was looking for.

Just curious, can you point me to some reference material for how you came to these expressions?  Especially the one for the numerator.

Thx!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Limit of Quotient of Infinite Series

Hi jhomme;

First things first, for some values of p and g, both a and b will not exist.

For those summations I used Mathematica. Since you wanted the limit, I tried to get to it as fast as possible.

a, is not responding to simple methods but b can be done using the geometric series.

Multiply both sides by the common ratio. ( 1+ g ).

Subtract 1 from 2.

---

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: Limit of Quotient of Infinite Series

Hi all,

First, let's find a.

Last edited by gAr (2011-06-02 17:46:55)

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"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."

jhomme

Member

Registered: 2011-06-02

Posts: 3

Re: Limit of Quotient of Infinite Series

Brilliant!  Thanks to both bobbym and gAr!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Limit of Quotient of Infinite Series

Your welcome! There is some nice stuff in gAr's solution. Interesting.

---

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: Limit of Quotient of Infinite Series

Hi jhomme,

You're welcome!

---

"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."