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EvLer
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Hi, I came across this puzzle, see if you can solve it :
[tex]\sqrt{1 + \sqrt{1 + 2\sqrt{1 + 3\sqrt{...}}}} = ?[/tex]
[tex]\sqrt{1 + \sqrt{1 + 2\sqrt{1 + 3\sqrt{...}}}} = ?[/tex]
Its one of those *aha* moments. Personally, these sort of "functional equations" are good fun to play around with (i have played with this one, quite a while back). Just remember the trick i used above and try to come up with some more radicals just like those.EvLer said:ahhh, i see... how did you see that?
have a link? or a reference?TenaliRaman said:There is a paper by T Vijayaraghavan (IIRC) which discussed convergence issues of such radicals, which could probably be the starting point of such analysis.
How would you check that the recursion is valid? EDIT: actually, since each step in the recursion produces the same number, why would the infinite recursion be any different? If it is different, what we have to do to check convergence?TenaliRaman said:Note in all of the above, its important to analyse that the infinite recursion is valid and that the infinite recursion actually converges.
When i said recursion being valid, i meant, making sure that the final number doesn't change by infinite application of recursion.AKG said:How would you check that the recursion is valid?
umm, let me try and give an example,actually, since each step in the recursion produces the same number, why would the infinite recursion be any different?
I am not sure whether there is a convergence check (the paper i mentioned above, discusses convergence of infinite radicals IIRC and not exactly recursion, and to be honest, when i had read that paper, i couldn't understand a penny out of it, :uhh: cmon i am just 21 and an engineer at that so give me a break ) .If it is different, what we have to do to check convergence?
If i had, i would have definitely linked to it in the original post. I don't even remember where i had read it, it was quite a long time back. Hell i was a toddler in maths when i had read that, i was just able to quote that paper because i had read that in reference to Srinivasa Ramanujan, who btw was the brains behind such infinite radicals.EvLer said:have a link? or a reference?
Let R(k,f(x)) denote the value of recursive function f(x) at depth k.
Then a possible convergence condition would be,
R(1,f(x)) = R(2,f(x)) = R(k,f(x)) = R(k+1,f(x))
Indeed!Hurkyl said:Why not apply the usual condition for convergence of a sequence? :tongue2:
I.E. does this limit exist?
[tex]
\lim_{k \rightarrow \infty} R(k, f)
[/tex]
A square root is a mathematical operation that calculates the number which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because 5 multiplied by 5 is 25.
The most common method for finding the square root of a number is by using a calculator or a computer. However, you can also estimate the square root of a number by using long division, or by using a method called the "guess and check" method.
A perfect square is a number that has an integer as its square root. For example, 25 is a perfect square because its square root is 5, which is an integer. On the other hand, 20 is not a perfect square because its square root is a decimal number (approximately 4.472).
Yes, negative numbers can have square roots. However, in order to find the square root of a negative number, you need to use imaginary numbers, which are numbers that involve the square root of -1 (also known as "i").
A square root is the inverse operation of squaring a number, while an exponent is a mathematical operation that indicates how many times a number is multiplied by itself. For example, the square root of 16 is 4 because 4 multiplied by itself (4 squared) equals 16. An exponent of 2, or 2 squared, also equals 16.