- #1

Stephen Tashi

Science Advisor

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## Main Question or Discussion Point

In the thread https://www.physicsforums.com/threads/recursive-square-root-inside-square-root-problem.954655/ a sequence interpreted from the notation:

##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}##

was discussed.

What sequence would you (fellow forum members) associate with that expression?

Define ##\{a_i\}## to be the sequence whose ##i## th term is a truncation of that expression after the ##i##th square root. So :

##a_1 = \sqrt{2}##

##a_2 = \sqrt{2 +\pi \sqrt{3 }}##

##a_3 = \sqrt{2 + \pi \sqrt{3 + \pi \sqrt{4}}}##.

etc.

I think of sequence ##\{a_i\}## and its limit as a safe and reliable context for discussing the meaning of the infinitely nested square roots, but others think of a different sequence.

This is my interpretation of the other sequence:

Define ##B = lim_{i \rightarrow \infty} a_i ##, assuming such a limit exists.

Define the sequence ##\{b_i\}## by ##b_1 = B## and the recurrence relation ##b_{n+1} = (b_n^2 - (n+1))/ \pi## .

So

## b_2 = ( B^2 - 2)/ \pi ## and we might denote it by ## \sqrt { 3 + \pi \sqrt{4 + ...}}##.

##b_3 = (b_2^2 - 3)/ \pi ## denoted by ## \sqrt{4 + \pi { \sqrt{5} + ...}}##.

etc.

There can be ambiguities in interpreting the "..." notation. (https://www.physicsforums.com/threads/ambiguities-of-the-notation.955191/#post-6055289 ) Are the sequences ##\{a_i\}## and ##\{b_i\}## the only reasonable seqeuences to use for assigning a value to the above expression of infinitely nested radicals?

##\large{\sqrt{2+\pi \sqrt{3+\pi\sqrt{4+\pi\sqrt{5+\dotsb}}}}}##

was discussed.

What sequence would you (fellow forum members) associate with that expression?

Define ##\{a_i\}## to be the sequence whose ##i## th term is a truncation of that expression after the ##i##th square root. So :

##a_1 = \sqrt{2}##

##a_2 = \sqrt{2 +\pi \sqrt{3 }}##

##a_3 = \sqrt{2 + \pi \sqrt{3 + \pi \sqrt{4}}}##.

etc.

I think of sequence ##\{a_i\}## and its limit as a safe and reliable context for discussing the meaning of the infinitely nested square roots, but others think of a different sequence.

This is my interpretation of the other sequence:

Define ##B = lim_{i \rightarrow \infty} a_i ##, assuming such a limit exists.

Define the sequence ##\{b_i\}## by ##b_1 = B## and the recurrence relation ##b_{n+1} = (b_n^2 - (n+1))/ \pi## .

So

## b_2 = ( B^2 - 2)/ \pi ## and we might denote it by ## \sqrt { 3 + \pi \sqrt{4 + ...}}##.

##b_3 = (b_2^2 - 3)/ \pi ## denoted by ## \sqrt{4 + \pi { \sqrt{5} + ...}}##.

etc.

There can be ambiguities in interpreting the "..." notation. (https://www.physicsforums.com/threads/ambiguities-of-the-notation.955191/#post-6055289 ) Are the sequences ##\{a_i\}## and ##\{b_i\}## the only reasonable seqeuences to use for assigning a value to the above expression of infinitely nested radicals?