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**Question****Guest**

What is the sum of the following.

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

**irspow****Member**- Registered: 2005-11-24
- Posts: 457

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

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

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

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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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Great!

Siva is right!

Well done, Siva!

IPBLE: Increasing Performance By Lowering Expectations.

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

thanks, krassi_holmz

glad to help

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

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**irspow****Member**- Registered: 2005-11-24
- Posts: 457

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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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

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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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,608

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

So perhaps the answer **is** The Golden Mean.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

I forgot to put the url back:

http://mathworld.wolfram.com/GoldenRatio.html

IPBLE: Increasing Performance By Lowering Expectations.

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

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

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

He'll call back.

IPBLE: Increasing Performance By Lowering Expectations.

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

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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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

"thank us" ???

IPBLE: Increasing Performance By Lowering Expectations.

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

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

Why did the vector cross the road?

It wanted to be normal.

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

Just a joke.

IPBLE: Increasing Performance By Lowering Expectations.

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

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

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

Nope, definitely not

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

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

"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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