Square Root Problem
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May 9th, 2016 at 2:16:58 PM
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What is the
Sqrt ( 1 + 2 * Sqrt (1 + 3 * Sqrt (1 + 4 * Sqrt (1 + .....)))) the nested square root going ad infinitum continuning with numbers 5,6 etc
Sqrt ( 1 + 2 * Sqrt (1 + 3 * Sqrt (1 + 4 * Sqrt (1 + .....)))) the nested square root going ad infinitum continuning with numbers 5,6 etc
May 9th, 2016 at 4:30:11 PM
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I admit I cheated and used Excel. That said...
My answer is 3, since you started the series with a 2.
No elegant solution yet.
My answer is 3, since you started the series with a 2.
No elegant solution yet.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
May 9th, 2016 at 4:53:32 PM
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I'm posting here to protect Wiz's answer. Apparently the spoiler doesn't hide in the latest forum post preview.
May 9th, 2016 at 5:02:42 PM
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I think I've seen something like this before. Its a special case of a relationship that Ramanujan discovered.
Basically,
x+1 = Sqrt (1+x* ( Sqrt (1+(x+1)* Sqrt ( 1+ (x+2)* Sqrt ( 1+ (x+3)* Sqrt ( 1+ .. . . .)
since x =2 in your question, this infinite series of nested radicals converges to x+1 = 3.
Basically,
x+1 = Sqrt (1+x* ( Sqrt (1+(x+1)* Sqrt ( 1+ (x+2)* Sqrt ( 1+ (x+3)* Sqrt ( 1+ .. . . .)
since x =2 in your question, this infinite series of nested radicals converges to x+1 = 3.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
May 9th, 2016 at 10:38:39 PM
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At x = 0, the thing is 1.
For very large x, the 1's and added 1's don't matter, and the continued root of x, itself, amounts to x. (Try it with sqrt(2sqrt(2sqrt(2sqrt(2)) in your browser, adding the "2sqrt(" part over and over beside the last 2.) So no fraction on the x's part.
No exponent on the x's part of the answer because with negative x, the thing would still be montonocally increasing.
Put the 1 with the x to get the thing at x+1. (And hope for the best. Lol.)
For very large x, the 1's and added 1's don't matter, and the continued root of x, itself, amounts to x. (Try it with sqrt(2sqrt(2sqrt(2sqrt(2)) in your browser, adding the "2sqrt(" part over and over beside the last 2.) So no fraction on the x's part.
No exponent on the x's part of the answer because with negative x, the thing would still be montonocally increasing.
Put the 1 with the x to get the thing at x+1. (And hope for the best. Lol.)
So much bullshit; so little time!
