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#1 2012-08-12 02:29:16

Especi
Member
Registered: 2012-06-29
Posts: 11

Help with Ibn Al Haytham recursive relationship.

Hello everyone , this is the relation I am given. I am required to prove this either algebraically or with the help of diagram of areas in a subdivided rectangle( I have found this on the internet). I am a bit stuck at the algebraic part, the methods I am trying at the moment to prove this are by induction and by changing the order of the summation on the RHS but I seem to have made a mistake as I have not reached the desired result. Could anyone assist me in solving this ?:)

Thank you very much for your help.

Last edited by Especi (2012-08-12 05:35:09)

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#2 2012-08-12 05:42:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: Help with Ibn Al Haytham recursive relationship.

Hi Especi;

Welcome to the forum. Can I see what you have done?


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#3 2012-08-12 07:39:24

Especi
Member
Registered: 2012-06-29
Posts: 11

Re: Help with Ibn Al Haytham recursive relationship.

I'll write briefly what I tried for this though I am not sure how correct it will be.

I started spliting the summation

to
. I then wrote
. I made  the notation
, substituted that in and tried prove the equality.

When I tried to prove it by induction I assumed that the relation is true for

and tried to prove it for
. I made use of some of the work I have done above but sadly nothing really came out of it.

I'm quite sure I have made several mistakes and if you would be so kind as to point them out that would be great smile . Even more annoying is that in the places I have looked for explanations mention that this is relatively easy to do so there must be something really obvious that I keep missing.

Thank you again for your help.

Last edited by Especi (2012-08-12 09:46:23)

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#4 2012-08-12 08:16:43

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: Help with Ibn Al Haytham recursive relationship.

Hi;

This step here is not correct:

looks like a typo.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#5 2012-08-12 09:47:36

Especi
Member
Registered: 2012-06-29
Posts: 11

Re: Help with Ibn Al Haytham recursive relationship.

That is correct it should be k , my mistake.

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#6 2012-08-12 12:02:06

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: Help with Ibn Al Haytham recursive relationship.

Hi;

Generally the above is not true. To sum a product you must use summation by parts or Abel summation.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#7 2012-08-12 13:09:56

cmowla
Member
Registered: 2012-06-14
Posts: 57

Re: Help with Ibn Al Haytham recursive relationship.

Ah, this wasn't too bad using algebra.













Q.E.D.

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#8 2012-08-12 13:28:20

Especi
Member
Registered: 2012-06-29
Posts: 11

Re: Help with Ibn Al Haytham recursive relationship.

I should have seen that this does not hold up for this case , I did come across the Able summation but I assumed that this would require something simpler. Could you possibly indicate a place to start from which I could prove the relation ?

Thank you for your help.

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#9 2012-08-12 13:58:50

cmowla
Member
Registered: 2012-06-14
Posts: 57

Re: Help with Ibn Al Haytham recursive relationship.

Especi wrote:

I should have seen that this does not hold up for this case , I did come across the Able summation but I assumed that this would require something simpler. Could you possibly indicate a place to start from which I could prove the relation ?

Thank you for your help.

What more is there to prove?  Did you not see my post?

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#10 2012-08-12 14:29:50

Especi
Member
Registered: 2012-06-29
Posts: 11

Re: Help with Ibn Al Haytham recursive relationship.

For some reason it did not show up , thank very much for your help.

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#11 2012-08-12 14:58:50

cmowla
Member
Registered: 2012-06-14
Posts: 57

Re: Help with Ibn Al Haytham recursive relationship.

Especi wrote:

For some reason it did not show up , thank very much for your help.

Oh, okay.  You're very welcome.  If there is any part of it which you need clarification, let me know...I didn't really write out any of the proof in English.:)

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#12 2012-08-12 16:38:07

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: Help with Ibn Al Haytham recursive relationship.

Hi Especi;

I did come across the Able summation

To increase your general knowledge of summations take a look at summation by parts too.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#13 2012-08-12 19:08:27

cmowla
Member
Registered: 2012-06-14
Posts: 57

Re: Help with Ibn Al Haytham recursive relationship.

bobbym wrote:

Hi Especi;

I did come across the Able summation

To increase your general knowledge of summations take a look at summation by parts too.

I am not familiar with that, so I looked it up here.
http://www.proofwiki.org/wiki/Summation_by_Parts


I held myself from looking at their proof so that I could prove it on my own in my uncondensed language and extra steps.  Here's my proof.

Statement:











If the statement I proved in this post is what you were referring to, how does it increase knowledge of series more than the process I used to prove the recurrence relationship in my previous post?

Is it because in the line:

we don't include

and therefore we don't have
at the nth term?

(When I grouped common f factors of g in the step of the proof written a second time above, for the last (nth) sum pair, only the first (positive) term in the pair exists.)

In other words, are you implying that this shows the basic idea of telescoping sums from basic Calculus II?

Last edited by cmowla (2012-08-12 19:13:29)

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#14 2012-08-12 19:19:28

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 90,663

Re: Help with Ibn Al Haytham recursive relationship.

cmowla wrote:

If the statement I proved in this post just now is what you were referring to

I suggested he look at SBP because in his proof he used the following idea


The above relation is not true. There are two general ideas to sum a product. SBP and Abel's formula. He said he came across Abel's already but SBP is more widely used and should be learned first.

, how does it increase knowledge of series more than the process I used to prove the recurrence relationship in my previous post?

I do not recall saying which increased his knowledge more. This is what I wrote.

bobbym wrote:

To increase your general knowledge of summations take a look at summation by parts too.

Since he was making a mistake on an exact type of SBP problem I thought that would help.


In mathematics, you don't understand things. You just get used to them.

I agree with you regarding the satisfaction and importance of actually computing some numbers. I can't tell you how often I see time and money wasted because someone didn't bother to run the numbers.

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#15 2012-08-12 19:33:33

cmowla
Member
Registered: 2012-06-14
Posts: 57

Re: Help with Ibn Al Haytham recursive relationship.

bobbym wrote:

I do not recall saying which increased his knowledge more. This is what I wrote.

bobbym wrote:

To increase your general knowledge of summations take a look at summation by parts too.

Since he was making a mistake on an exact type of SBP problem I thought he might like to look it up.

Oh.  That was actually the first thing that popped into my head, but it stayed in there for such a short time that I simply forgot.:)

I have sadly never heard of Abel's formula until today, but it definitely shows that the sum of products is not equal to the product of the sums.  Great point.  I'll have to take a closer look at the formula.

I think something that might throw off a lot of students is the fact that:

, but I could be wrong.

Last edited by cmowla (2012-08-12 19:34:12)

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