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#1 2005-12-30 07:38:38

Question
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Help

What is the sum of the following.

squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...

#2 2005-12-30 07:50:18

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Help

There is no sum since the series diverges.  Conceptually it approaches infinity.

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#3 2005-12-30 07:52:16

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Re: Help

Let

squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...) = x

[squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...)]^2 = x^2

(1+(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...) = x^2

(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...) = x^2 - 1

As we already said,  squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(squr(1+(...) = x

so

x = x^2 -1

x^2 - x - 1 = 0

From the quadratic formula

x = (-(-1) + squr( 1 - 4*1*(-1)))/(2*1) or x = (-(-1) - squr( 1 - 4*1*(-1)))/(2*1)

x = (1+ squr (5))/2 or  x = (1 - squr (5))/2

x = 1.618 or x = -0.618

x = -0.618 won't work since it is negative number, so the sum is 1.618.

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#4 2005-12-30 08:04:36

Flowers4Carlos
Member
Registered: 2005-08-25
Posts: 106

Re: Help

siva:  that is a very interesting approach and one that i'm not too sure if it's true, however, it looks like you found the x-intercept and not the summation.

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#5 2005-12-30 08:19:07

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

Great!
Siva is right!
Well done, Siva!


IPBLE:  Increasing Performance By Lowering Expectations.

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#6 2005-12-30 08:19:32

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Re: Help

thanks, krassi_holmz
glad to help

Last edited by siva.eas (2005-12-30 08:19:51)

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#7 2005-12-30 08:31:29

irspow
Member
Registered: 2005-11-24
Posts: 455

Re: Help

siva.eas, your answer works if there is a final term in the sequence and that it is indeed 1.  But exactly how one would show that a final term exists at infinity is beyond me for this equation.

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#8 2005-12-30 09:04:44

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

Number of the form


is nested radical.
Herschfeld (1935) proved that a nested radical of real nonnegative  terms converges iff
is bounded

Last edited by krassi_holmz (2005-12-30 09:13:42)


IPBLE:  Increasing Performance By Lowering Expectations.

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#9 2005-12-30 09:09:12

MathsIsFun
Administrator
Registered: 2005-01-21
Posts: 7,552

Re: Help

I tried it in Excel, and it reaches 1.618034 after 20 terms ...

So perhaps the answer is The Golden Mean.


"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

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#10 2005-12-30 09:34:53

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

I forgot to put the url back:
http://mathworld.wolfram.com/GoldenRatio.html


IPBLE:  Increasing Performance By Lowering Expectations.

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#11 2005-12-30 13:31:31

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Re: Help

Funny, we never heard back from the person who asked the question.

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#12 2005-12-30 21:39:27

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

He'll call back.


IPBLE:  Increasing Performance By Lowering Expectations.

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#13 2005-12-31 08:21:09

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

Re: Help

Unless he's rude, in which case he'll look at the answer and not bother to thank us. It's been known to happen.


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

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#14 2005-12-31 09:23:40

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

"thank us" ???
smile smile smile smile smile


IPBLE:  Increasing Performance By Lowering Expectations.

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#15 2005-12-31 09:26:20

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

Re: Help

Well, us as a forum community. You knew what I meant. smile


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

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#16 2005-12-31 10:19:34

krassi_holmz
Real Member
Registered: 2005-12-02
Posts: 1,908

Re: Help

Just a joke.


IPBLE:  Increasing Performance By Lowering Expectations.

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#17 2005-12-31 12:09:45

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Re: Help

I do not think he is ever goint to reply back. It has been about 30 hours now.

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#18 2006-01-02 13:51:14

siva.eas
Member
Registered: 2005-09-17
Posts: 166

Re: Help

Nope, definitely not

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#19 2006-01-02 15:04:59

Ricky
Moderator
Registered: 2005-12-04
Posts: 3,791

Re: Help

Heh, 30 hours?  Jeez, talk about no patience... wink


"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

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