Let $$S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$$ Is it possible to write $S$ in terms of standard mathematical functions and operators? If yes, what is the exact value of $S$? If not, can it be proved?
2026-03-25 15:42:26.1774453346
Find the exact value of expression $S=\sqrt{4+\sqrt[3]{4+\sqrt[4]{4+\sqrt[5]{4+\sqrt[6]{4+\cdots}}}}}$
239 Views Asked by user164524 https://math.techqa.club/user/user164524/detail AtRelated Questions in CLOSED-FORM
- How can I sum the series $e^{-2}\frac{(3)^n}{n!}\sum_{k=0}^{\infty}\left ( \frac{1}{2}\right )^k\frac{1}{(k-n)!}$
- Computing $\int_0^\pi \frac{dx}{1+a^2\cos^2(x)}$
- Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?
- Finding a closed form for a simple product
- For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?
- Convergence of $\ln\frac{x}{\ln\frac{x}{\ln x...}}$
- How can one show that $\int_{0}^{1}{x\ln{x}\ln(1-x^2)\over \sqrt{1-x^2}}\mathrm dx=4-{\pi^2\over 4}-\ln{4}?$
- Exercises about closed form formula of recursive sequence.
- Simplify and determine a closed form for a nested summation
- Direction in closed form of recurrence relation
Related Questions in NESTED-RADICALS
- A sequence of nested radicals and its limit
- Prove that this formulae equals 3
- Alternate proof for Viète's infinite product of nested radicals
- Is what I have got about $\int\sqrt{x+\left(\sqrt{{x}+\sqrt{x+\cdots}}\right)}dx$ true or false?
- Simplification via Denesting Radicals
- Proof that $\sqrt{1+2\sqrt{1+2\sqrt{1+2\sqrt{1+2(\ldots)}}}} = 1+\sqrt{2}$
- Exclusive way to solve irrational equation
- What is the value of $\sqrt{1+ \sqrt[3]{1+\sqrt[4]{1+ \sqrt[5]{1+ \cdots }}}}$?
- Simplification of radicals: $\sqrt{4+\sqrt5}-\sqrt{\frac{4+\sqrt{11}}{2}}$
- Simplifying nested radicals? $\left(\sqrt{4+\sqrt{16-a^2}}+\sqrt{4-\sqrt{16-a^2}}\right)^2$
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?