[math-fun] How do you solve this kind of problem?
Question -- I've seen many problems of the kind << Express in closed form the number x represented by x = sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n + . . . . . . ) . . . )))
And when I numerically take the limit of sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n) . . . ))) as n -> oo (using 16-digit precision) the answer converges to 1.7579327566180045 by the time n = 20. But how would one identify this number? And what about the many closely related problems? (For example, if in the quoted problem the integers 1,2,3,... are replaced by their squares, the answer numerically (again to 16 places) converges to 1.9426554227639874 even sooner -- by the time n = 15. In fact, I wonder if there's a way to express in closed form the limit when the integers 1,2,3,... are replaced by their pth powers (p an arbitrary real) ? Call this limit rad(p). Numerically, I was quite surprised to see what limit as p -> 0+ of rad(p) is (though I shouldn't have been.) --Dan --Dan
----- Original Message ----- From: "Daniel Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, November 14, 2006 19:30 Subject: [math-fun] How do you solve this kind of problem?
Question -- I've seen many problems of the kind
<< Express in closed form the number x represented by
x = sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n + . . . . . . ) . . . )))
And when I numerically take the limit of
sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n) . . . ))) as n -> oo
(using 16-digit precision) the answer converges to 1.7579327566180045 by the time n = 20.
But how would one identify this number?
You might want to look at the references mentioned at <http://www.research.att.com/~njas/sequences/A072449>. David W. Cantrell
And what about the many closely related problems?
(For example, if in the quoted problem the integers 1,2,3,... are replaced by their squares, the answer numerically (again to 16 places) converges to 1.9426554227639874 even sooner -- by the time n = 15.
In fact, I wonder if there's a way to express in closed form the limit when the integers 1,2,3,... are replaced by their pth powers (p an arbitrary real) ?
Call this limit rad(p).
Numerically, I was quite surprised to see what
limit as p -> 0+ of rad(p) is
(though I shouldn't have been.)
--Dan
Is the `obviously' comment at A072449 correct? R. On Tue, 14 Nov 2006, David W. Cantrell wrote:
----- Original Message ----- From: "Daniel Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Tuesday, November 14, 2006 19:30 Subject: [math-fun] How do you solve this kind of problem?
Question -- I've seen many problems of the kind
<< Express in closed form the number x represented by
x = sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n + . . . . . . ) . . . )))
And when I numerically take the limit of
sqrt(1 + sqrt(2 + sqrt(3 + . . . + sqrt(n) . . . ))) as n -> oo
(using 16-digit precision) the answer converges to 1.7579327566180045 by the time n = 20.
But how would one identify this number?
You might want to look at the references mentioned at
<http://www.research.att.com/~njas/sequences/A072449>.
David W. Cantrell
And what about the many closely related problems?
(For example, if in the quoted problem the integers 1,2,3,... are replaced by their squares, the answer numerically (again to 16 places) converges to 1.9426554227639874 even sooner -- by the time n = 15.
In fact, I wonder if there's a way to express in closed form the limit when the integers 1,2,3,... are replaced by their pth powers (p an arbitrary real) ?
Call this limit rad(p).
Numerically, I was quite surprised to see what
limit as p -> 0+ of rad(p) is
(though I shouldn't have been.)
--Dan
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participants (4)
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Daniel Asimov -
David W. Cantrell -
Gareth McCaughan -
Richard Guy