The given problem involves finding a closed-form expression for the infinite nested radical (\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\dots}}}).

First, we denote the entire expression as (x). This leads to the recursive definition where each term (a_n) (starting with the (n)-th root) is given by (a_n = \sqrt[n]{4 + a_{n+1}}), with (x = a_2). The recurrence relation derived is (a_n^n = 4 + a_{n...

Let x denote the value of the nested radical in the title: x = sqrt(4 + ³√(4 + ⁴√(4 + …))).

If we define the tail recursively by a_k = k-th root of (4 + a_{k+1}), for k = 2,3,… then x = a_2 and the recurrence is a_k^k = 4 + a_{k+1}. Unlike the multiplicative case you wrote (which is linear after taking logs and gives 4^{e−2}), this additive nesting is genuinely nonlinear: ln a_k = (1/k) ln(4 + e...

I'll analyze this nested radical problem step by step.

The expression in question is:

Let me denote this value as and work backwards from the known result about .

From the given information, we know that:

Let me define a more general nested radical:

This is a fascinating question that touches on some of the deepest open problems in number theory. The short answer is:

It is strongly conjectured that no such infinite sequence exists, but this remains an open problem.

No one has been able to prove that every factonomial sequence must terminate, but there are several powerful arguments that suggest this is the case. Since you're open to ans...

<think> I need to find a closed-form expression for \(\sqrt{4 + \sqrt[3]{4 + \sqrt[4]{4 + \dots}}}\). The problem references an expression for \(4^{e-2}\) which is similar but with multiplication instead of addition. That expression is \(4^{e-2} = \sqrt{4 \cdot \sqrt[3]{4 \cdot \sqrt[4]{4 \cdots}}}\).

For the addition version, I recall that expressions like (\sqrt{a + \sqrt{a + \sqrt{a + \cdots}...

A convenient way to see why there really is no “nice” finite‐algebraic closed‐form for

  x = √(4 + ³√(4 + ⁴√(4 + ⋯)))

is to rewrite it as an infinite product or an infinite series. In particular set

  x₁ = x,
  x_{k} = (4 + x_{k+1})^{1/(k+1)}, k = 1,2,3,…,

and note that x_{k} →1 as k→∞. Then for each k

  x_{k} = 4^{1/(k+1)}·(1 + x_{k+1}/4)^{1/(k+1)}

so

(1) x = x₁
 = ∏_{k=...

Write

      x2  = √(4+ x3 ),  
      x3  = ³√(4+ x4 ),  
      x4  = ⁴√(4+ x5 ), … ,               (1)

and denote by

      S = x2 = √{4+√[3]{4+√[4]{4+… }}}                         (2)

the value that is asked for.

1. Convergence

For n ≥ 2 put fn (t) = (4+t)1/n .
Because fn is continuous and strictly increasing on [0,∞) and

      1 < fn (t) ≤ 4 1/n + t 1/n  ≤ 4 1/...

Let

R = √( 4 + ³√( 4 + ⁿ√( 4 + … ) ) ). (1)

More formally, put

uN = N√(4)
u_{k} = k√( 4 + u_{k+1} ) (k = N−1,N−2,…,2) (2)

and define

R := lim_{N→∞} u₂ . (3)

1. Convergence and first estimates
   ———————————————————————
   Because k√x is increasing in x and smaller than max{...

Finding a Closed Form for

I'll work on finding a closed-form expression for the nested radical in the title.

First, let's understand what we already know from the problem statement:

• We have
• This gives us

Now, I...