What are the answers to this series?

We need to be more precise in terms of what exactly we are looking for. It does not make sense to say "What are the solutions to the series √[ 1 - √( 17/16 - √[ 1 - √( 17/16 - √[ 1 - √( 17/16 - √[ 1 -√( 17/16 ... ) ] ) ] ) ] ) ]," since the assumption that there *is* a solution, i.e. a well-defined value of the continued radical, means that such a solution must be unique (if a sequence converges, it must converge to a unique value).

It *is* true that, if x = √[ 1 - √( 17/16 - √[ 1 - √( 17/16 - √[ 1 - √( 17/16 - √[ 1 -√( 17/16 ... ) ] ) ] ) ] ) ] is well-defined, that x satisfies x = √[ 1 - √( 17/16 - x)]. In the process of solving this equation, we obtain the intermediate equations:

x² = 1 - √(17/16 - x)

(x² - 1)² = [-√(17/16 - x)]²

x⁴ - 2x² + x - 1/16 = 0

The final equation does indeed have four real solutions: x = 1/2 and x ≈ 0.0731827, which you mention, as well as x ≈ -1.62457 and x ≈ 1.05139 [1]. However, the latter two solutions of the auxiliary equation are easily seen to not satisfy the original defining equation of x, as x ≈ -1.62457 is real but forces an imaginary value of √(1 - √(17/16 - x)), and it can also be verified directly that x ≈ 1.05139 is also an extraneous solution [2]. Hence, *the* solution to the problem, if it exists, is one of 1/2 or ≈ 0.0731827.

My instinct is that the continued radical is well-defined and equal to 1/2, but I don't have the time right now to prove that properly, perhaps later this evening.

Edit: Scythian, saying a sequence converges at all implies that it converges to a single value -- that follows immediately from the definition of convergence. We don't say that the series 1 + -1 + 1 + -1 + ... converges to "either 0 or 1, depending on what the 'last term' is," because there is no "last term." We simply say that it does not converge. You are right that convergence questions of this type can be tricky. It is reasonable to say that the continued radical is equal to the limit of the recursive sequence a_(n + 1) = √(1 - √[17/16 - a_n]), but we do not a priori know that convergence of this sequence is not dependent upon the choice of initial condition a0. It is possible that different a0 lead to different limits, which is what makes developing an analytical proof of convergence to 1/2 relatively difficult.

There are results on nested radicals of this type: namely, the Herschfield convergence theorem [3] establishes that, given a sequence {a_n} of nonnegative terms, the continued radical √[a1 + √(a2 + √[a3 + ... converges to a unique value so long as {a_n}^(1/2^n) is bounded. Unfortunately, we can't apply that here, because of the presence of negative signs in the nested radical, but we can see numerically that the recursive sequence a_n = √(1 - √[17/16 - a_(n - 1)]) seems to converge to 1/2 (or 1/2 + 0j, if complex a_n are permitted -- convergence of the result does not seem to be limited to real sequences) regardless of the choice of initial a0 (Python script at [4]).

Edit 2: I spoke too soon, and said something foolish. Obviously, a_n cannot converge to 1/2 for all values, because the value x ≈ 0.073187 is a fixed point of the recurrence relation that defines a_n, and so a choice of a0 = x must result in a_n = x for n > 0. But what is interesting is the fact that in my approximation script, using absurdly small values of epsilon and a value of a0 numerically correct to with one part in 10^15, a_n *still* converges numerically to 1/2. Thus, it seems that x ≈ 0.073187 is *not* an attractive fixed point: in fact, it might be the case that a_n converges to 1/2 for all a0 not equal to that x. If x *is* an attractive fixed point (on some extremely tiny δ-neighborhood of x), I'd need more machine precision to see evidence of it.

What does this mean? I'm not entirely certain: do we conclude that the nested radical is ill-defined because different a0 lead to different limits? Do we declare it equal to 1/2, since that is the limit almost always? Is x ≈ 0.073187 is actually a solution to the equation, or just ridiculously close to one? I don't really feel like pushing through the Cardano formula to get a closed form solution, and even Wolfram doesn't give a closed form solution for two of the roots of 8x³ + 4x² -14x + 1 (including the one we need, unfortunately).

My personal take is as follows: since 1 and 17/16 are the only two terms inside the nested radical, it is sensible to take its value as lim a_n, with a0 = 1 or 17/16 (the two are equivalent, as a0 = 17/16 --> a1 = 1), in which case the radical is well-defined and equal to 1/2. I'm in no mood to prove that, but the numerical evidence is overwhelming. The other "solutions" to the problem are nothing but artifacts that arise in determining the actual solution: the statement that x = √(1 - √(17/16 - √1 -... implies that x satisfies x = √(1 - √(17/16 - x)), but the converse needn't be true.

Well, right off it can be quickly be estimated to be 1/2. Which is also one (and only real) root of the equation x = √(1 - √( 17/16 - x)). The other two are complex.

To estimate, guess a number, plug it in at the end of the series you have posted (with enough terms), and see if you get it right back. In other words, we're trying to find zeros of x - S(x) = 0.

Unless the brackets [ ] means something other than ( ).

Edit: I correct myself. The "other two" are not complex. There is one more real value, which is 0.0731827445. NOW I give this problem a TU. Worth another look.

Edit 2: I'm having problems with those two "other complex roots", because they don't behave like zeros. The equation to be solved is a quartic, so that there should be 4 roots, of which we know that 2 are real, and the other 2 seem to be complex. So, you say all 4 are real?

Edit 3: ejwaxx, this kind of nested radical problem always seem to lead into arguments about "what it really converges to" Consider the simple series: 1 - 1 + 1 - 1 + 1 .... What does it converge to? It can be either 0 or 1, depending what that "last term" is. The problem is, that "last term" is not unique nor well defined. Likewise, convergence of nested radicals like this isn't well defined nor necessarily unique either. Sometimes it can be, other times not. I am looking forward to your explanation on how to show that convergence of such nested radicals can be well defined and unique, or at least explain why 1/2 is the "only correct" answer to FGR!'s problem.

1) could you tell me, in one way or another, how many, if any, questions, if you could call these questions, are in this series, if this is a series, of questions? - idk.....4? 2) How did you go at understanding, if you understood, the last question, if the last question could be called a question in this disputable series of questions? -ummmmm.........I guess it was aking how many questions are here.....idk I'm confused 3) what is more annoying, a door that has been left ajar, not quite closed, not quite open, a crack of light showing through, or, a window with a crack, not a but* crack, but a crack in the glass, not the drug crack, but a crack in the glass?....crack in the glass Q4) how irritating, if irritating, and how much sense, if they make sense, do these questions make, answer on a scale, not a scale that weighs, but a scale, a score, of 1-10. irritating: 10 Sense: 4 Q4) Thankyou, if i should be thanking you, for being a willing, if you are willing, participant, if you are a participant, in this series, if it is a series, of qeustions? Well it sure beat doing ABS work....

Hint:

Let the left hand side be "a" and write it as a= √[1-b]

a^2=1-b and we can write b^2=17/16-a

combining we can get an equation that has .5 as a solution....

If you iterate x ----> √[ 1 - √( 17/16 - x) ],

then if you start below the root 0.07... you fall sooner or later out of the range where the iteration makes sense namely [1/16, 17/16].

If you start above 0.07.. but below 17/16, then your process converges towards 1/2. So that's the only meaningful value to your iterated radical.

The proof is elementary, but the result can be seen in 2 secs by plotting the graph