Nested surds expression

Iderown

One for the board gurus here - hoping that you have seen it before and can remember the solution (if any!). It was given to me by a buddy on another internet forum. It is NOT a course homework or assessed problem. Purely for fun.

Does a neat, closed form formula (not one featuring an infinite composition of radicals!) exist for:



If so, what is it?

A truncated calculation seems to indicate that the numerical result is 1.51186... But, I'm not really sure about this. Even less sure about its infinite extension. I will explain why in a further post if necessary.

Would this type of problem be within the area of Mathematical Analysis?

Is there a neat trig substitution, for example, using

or

which could make the problem more tractable?

MathsManiac

Did you check out this at Wolfram?
http://mathworld.wolfram.com/NestedRadical.html

ray giraffe

Probably the expression for root(2) he was trying to give you, it's number (18) on the mathworld page, but it's not as simple as he thought. It is cool tho

Iderown

Thanks, guys.

I'll have a study of the item at Wolfram later today when I waken up a bit. On first reading, the numbers in expression (18) seem to "die" a bit faster than those above. But, worth a few hours study.

Iderown

The first result that I quoted (1.511865...) was got using the truncated formula above. Several truncations, using 25, 100 and 500 levels of nesting converged to this value from below. In the longer sets, most of the intermediate values were 1.

I did not have enough memory to do the infinite nesting.

A small bit of arithmetic on the original form - involving taking out factors from within each surd so that the leading number is 1 - results in an alternative form:



Numbers here, the (2 to power of) appear to be increasing left to right. Spreadsheet calculation involving 50 or so nestings results also in 1.511865 ... This time approaching the value from above.