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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: 1,055

Re: Help

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

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.

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.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Help

Great!
Siva is right!
Well done, Siva!

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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)

irspow

Member

Registered: 2005-11-24

Posts: 1,055

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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I am at an age where I have forgotten more than I remember, but I still pretend to know it all.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

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)

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IPBLE:  Increasing Performance By Lowering Expectations.

MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Help

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

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Help

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

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IPBLE:  Increasing Performance By Lowering Expectations.

siva.eas

Member

Registered: 2005-09-17

Posts: 166

Re: Help

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

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Help

He'll call back.

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IPBLE:  Increasing Performance By Lowering Expectations.

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.

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It wanted to be normal.

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Help

"thank us" ???

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IPBLE:  Increasing Performance By Lowering Expectations.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Help

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

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

krassi_holmz

Real Member

Registered: 2005-12-02

Posts: 1,905

Re: Help

Just a joke.

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IPBLE:  Increasing Performance By Lowering Expectations.

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.

siva.eas

Member

Registered: 2005-09-17

Posts: 166

Re: Help

Nope, definitely not

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Help

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

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"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..."