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Nested radical

A nested radical is a mathematical expression in which one or more radical signs contain additional radical expressions inside them, forming a recursive structure; common forms include finite expressions like \sqrt{a + \sqrt{b + \sqrt{c}}} and infinite ones like \sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \cdots}}}, where the a_i are typically positive real numbers. These expressions arise in algebra and analysis, with key areas of study encompassing their simplification (denesting), convergence for infinite cases, and exact evaluations that yield integers or irrational constants.^[1] The historical development of nested radicals traces back to the late 16th century, when François Viète employed infinite nested square roots to derive an approximation for \pi, expressed as \frac{2}{\pi} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{2 + \sqrt{2}}}{2} \cdot \frac{\sqrt{2 + \sqrt{2 + \sqrt{2}}}}{2} \cdots. Interest grew in the 19th century through the development of Galois theory, which addresses solvability of equations by radicals, and accelerated in the early 20th century with Srinivasa Ramanujan's identities, such as the infinite nested radical \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{\cdots}}}} = 3.^[2] Ramanujan's contributions, published in 1911, included general denesting formulas that simplify expressions like \sqrt{m + \sqrt{4m - 8n + n}} to non-nested forms involving sums of square roots.^[3] For infinite nested radicals of the form \sqrt{x_1 + \sqrt{x_2 + \sqrt{x_3 + \cdots}}} with nonnegative x_n, convergence is guaranteed if and only if \sup_n x_n^{1/2^n} < \infty, as proved by Aaron Herschfeld in his analysis of the limiting behavior of the associated sequence.^[4] Denesting finite nested radicals, particularly those of depth two, often reduces them to sums of simpler radicals; for instance, \sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}.^[1] Notable applications include exact representations of the golden ratio \phi = \frac{1 + \sqrt{5}}{2} = \sqrt{1 + \sqrt{1 + \sqrt{1 + \cdots}}} and further infinite products for \pi inspired by Viète's original formula.

Finite Nested Radicals

Denesting

Denesting is the process of simplifying a finite nested radical by expressing it in a form without nested roots, typically rewriting an expression like \sqrt{a + b\sqrt{c}} as \sqrt{d} + \sqrt{e}, where a, b, c, d, e are rational numbers and the square roots are no longer nested.^[5] This simplification reduces the depth of nesting and often facilitates further algebraic manipulation or computation.^[3] A key criterion for denestability applies to radicals of the form \sqrt{a + \sqrt{b}}, where a > 0 and a, b are rational: the expression can be denested if a^2 - b is the square of a rational number \delta > 0.^[5] In such cases, the denested form is \sqrt{\frac{a + \delta}{2}} + \sqrt{\frac{a - \delta}{2}}.^[6] For the more general \sqrt{a + b\sqrt{c}}, denestability holds if a^2 - b^2 c is a perfect square, allowing a similar decomposition assuming the form \sqrt{x} + \sqrt{y} and solving the resulting system x + y = a and $2\sqrt{xy} = b\sqrt{c}.^[6] Early methods for denesting radicals emerged in 16th- and 17th-century European algebra texts, particularly in the context of solving cubic equations where nested roots appeared in Cardano's formula, as developed by Tartaglia and Cardano.^[7] These techniques laid foundational approaches for handling radical expressions, though systematic denesting of square roots gained prominence in later algebraic treatises.^[8] A classic example is the simplification of \sqrt{3 + 2\sqrt{2}}. Rewrite $2\sqrt{2} = \sqrt{8}, yielding \sqrt{3 + \sqrt{8}}. Here, a = 3, b = 8, and a^2 - b = 9 - 8 = 1 = 1^2, a perfect square with \delta = 1. Thus,

\sqrt{3 + \sqrt{8}} = \sqrt{\frac{3 + 1}{2}} + \sqrt{\frac{3 - 1}{2}} = \sqrt{2} + \sqrt{1} = \sqrt{2} + 1.

To verify algebraically, square the proposed denested form:

(\sqrt{2} + 1)^2 = 2 + 2 \cdot 1 \cdot \sqrt{2} + 1 = 3 + 2\sqrt{2},

which matches the original radicand.^[6] This process relies on the fundamental identity

(\sqrt{x} + \sqrt{y})^2 = x + y + 2\sqrt{xy},

where equating coefficients derives the values of x and y.^[5] Such denesting techniques form the basis for more advanced identities, including those explored by Ramanujan for higher-degree nested radicals.^[9]

Two Nested Square Roots

A nested radical involving exactly two square roots takes the form \sqrt{a \pm \sqrt{c}}, where a and c are positive rational numbers. This expression can be denested into a sum or difference of two non-nested square roots under specific conditions. The denesting theorem states that if a > 0 and a^2 - c is the square of a positive rational number (i.e., a perfect square in the rationals), then

\sqrt{a \pm \sqrt{c}} = \sqrt{\frac{a + \sqrt{a^2 - c}}{2}} \pm \sqrt{\frac{a - \sqrt{a^2 - c}}{2}},

where the outer signs are chosen consistently with the original expression (plus for plus, minus for minus), and the resulting inner square roots are positive real numbers.^[5] The proof begins by assuming the denested form \sqrt{a \pm \sqrt{c}} = \sqrt{x} \pm \sqrt{y}, where x and y are positive rationals. Squaring both sides yields a \pm \sqrt{c} = x + y \pm 2\sqrt{xy}. Equating the rational parts gives x + y = a, and equating the coefficients of the irrational parts gives \pm 2\sqrt{xy} = \pm \sqrt{c}, so $4xy = c. Solving the system x + y = a and xy = c/4 leads to the quadratic equation t^2 - a t + c/4 = 0, with discriminant a^2 - c. For x and y to be rational, the discriminant must be a perfect square, say \delta^2 where \delta > 0 is rational. The solutions are then x = \frac{a + \delta}{2} and y = \frac{a - \delta}{2} (or vice versa, ensuring both are positive). Substituting \delta = \sqrt{a^2 - c} completes the derivation.^[5] The conditions for denestability are that a and c are positive rationals with \sqrt{c} irrational, a \pm \sqrt{c} > 0, and the discriminant a^2 - c is the square of a positive rational. If these hold, the expression simplifies to two separate square roots over the rationals; otherwise, it cannot be denested in this manner using only square roots.^[5] For example, consider \sqrt{4 + \sqrt{12}}. Here, a = 4 and c = 12, so a^2 - c = 16 - 12 = 4 = 2^2, a perfect square. Then \delta = 2, x = \frac{4 + 2}{2} = 3, and y = \frac{4 - 2}{2} = 1, yielding \sqrt{4 + \sqrt{12}} = \sqrt{3} + \sqrt{1} = 1 + \sqrt{3}. Verification: (1 + \sqrt{3})^2 = 1 + 2\sqrt{3} + 3 = 4 + 2\sqrt{3}, and since \sqrt{12} = 2\sqrt{3}, this matches $4 + \sqrt{12}. Similarly, for the minus case, \sqrt{4 - \sqrt{12}} = \sqrt{3} - 1, as (\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} = 4 - \sqrt{12}, and \sqrt{3} - 1 > 0.^[5]

Ramanujan's Identities

Srinivasa Ramanujan, in his mathematical notebooks compiled during the early 1900s, explored elegant identities expressing integers as finite nested radicals with increasing integer coefficients. These entries, later analyzed and published, highlight Ramanujan's profound insight into algebraic structures that simplify through pattern recognition and inductive methods.^[10] A prominent identity involves the finite nested radical

\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + \cdots + n\sqrt{1 + (n+1)(n+3)}}}} = 3,

where the nesting terminates with the non-radical term (n+1)(n+3) inside the last square root, generalizing to yield the integer 3 for any finite depth n \geq 1.^[10] The proof relies on inductive squaring and pattern recognition, leading to a telescoping verification. Assume the innermost expression $1 + (n+1)(n+3) = (n+2)^2, so its square root equals n+2. Squaring the previous level gives $1 + n(n+2) = 1 + n^2 + 2n = (n+1)^2, whose square root is n+1. Continuing outward, each level squares to (k+1)^2 = 1 + k(k+2) for decreasing k, telescoping down to the outermost \sqrt{1 + 2 \cdot 4} = \sqrt{9} = 3. This inductive process confirms the equality holds exactly for the finite structure.^[10] Additional examples illustrate the versatility. A shifted variant equals 4:

\sqrt{6 + 2\sqrt{7 + 3\sqrt{8 + 4 \cdot 7}}} = 4.

Here, the innermost $8 + 4 \cdot 7 = 36 = 6^2, so \sqrt{36} = 6; then $7 + 3 \cdot 6 = 25 = 5^2, \sqrt{25} = 5; finally $6 + 2 \cdot 5 = 16 = 4^2, \sqrt{16} = 4. This follows the same inductive squaring, starting from a base matching the pattern for 4.^[10] For the case n=3, the expression is

\sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4 \cdot 6}}}.

Innermost: $1 + 4 \cdot 6 = 25, \sqrt{25} = 5. Next: $1 + 3 \cdot 5 = 16, \sqrt{16} = 4. Outermost: $1 + 2 \cdot 4 = 9, \sqrt{9} = 3. Successive squaring verifies: $3^2 = 9 = 1 + 2 \cdot 4, $4^2 = 16 = 1 + 3 \cdot 5, $5^2 = 25 = 1 + 4 \cdot 6, confirming the equality through the telescoping pattern.^[10]

Landau's Algorithm

In 1989, Susan Landau developed a systematic algorithm to determine whether a finite nested radical expression over the rationals can be denested, meaning it can be rewritten using radicals of lower nesting depth while remaining in the same field extension.^[11] The algorithm leverages Galois theory to analyze the structure of the field extension generated by the nested radical, specifically by examining the minimal polynomial and its splitting field to identify subfields corresponding to simpler radical expressions.^[11] The core steps of Landau's algorithm begin with computing the minimal polynomial of the nested radical α over the rationals ℚ. This polynomial is derived recursively from the structure of the nested expression, resulting in a polynomial of degree 2^d where d is the nesting depth. Once obtained, the algorithm factors this minimal polynomial over algebraic extensions, checking for quadratic factors or intermediate subfields of degree 2 that would indicate denestability into square roots. If the Galois group of the splitting field admits a quadratic subextension containing α, a denested form exists; otherwise, the expression is irreducible in terms of lower-depth radicals. Irreducibility is tested using criteria such as Eisenstein's criterion applied to the minimal polynomial after suitable substitutions.^[11] The running time of the algorithm is exponential in the nesting depth d, due to the degree of the minimal polynomial growing as 2^d and the need to compute the full splitting field, which involves factoring polynomials over number fields of comparable degree. However, for fixed depth d, the time complexity is polynomial in the bit size of the coefficients of the input radicals, making it practical for shallow nestings.^[11] A representative application involves determining the denestability of the finite nested radical √(√8 + √(32 + √(80 + ⋯ + √(8n^2)))) for increasing n. Landau's algorithm computes the minimal polynomial iteratively for each added layer, revealing whether the overall extension factors into quadratics; for certain terminating depths, it confirms denestability into a form like (√m + √k)/l, while deeper nestings may remain irreducible.^[11] For the basic case of a double nested square root of the form α = √(a + b √c) with a, b, c ∈ ℚ, the minimal polynomial is given by

\begin{align*} x^4 - 2a x^2 + (a^2 - b^2 c) &= 0. \end{align*}

Denestability holds if this quartic factors into quadratics over ℚ, which occurs precisely when the resolvent cubic has a rational root, testable via the discriminant or direct factorization. Irreducibility tests, such as checking if the polynomial is Eisenstein at a prime dividing b but not c, confirm when no such denesting is possible.^[11]

Applications of Finite Nested Radicals

In Trigonometry

Finite nested radicals frequently appear in exact expressions for trigonometric functions evaluated at rational multiples of π, particularly when deriving values using multiple-angle formulas such as half-angle and triple-angle identities. These formulas allow the reduction of known trigonometric values for larger angles to expressions involving square roots for smaller angles, often resulting in nested structures before simplification. For example, repeated application of the half-angle formula for cosine, \cos(\theta/2) = \pm \sqrt{(1 + \cos \theta)/2}, generates nested radicals when iterated multiple times starting from a basic angle like π/3 or π/4. In the 16th century, mathematicians including François Viète employed these identities to compute exact trigonometric values, supporting the development of precise tables for navigation and astronomy. Viète's work on trigonometric relations and polygon-based approximations to π highlighted the utility of such expressions in obtaining closed-form solutions without numerical approximation.^[12] Specific examples illustrate this connection. The sine of π/5 (36°) is \sin(\pi/5) = \sqrt{(10 - 2\sqrt{5})/16}, derived from the multiple-angle formula for sin(5θ) applied to a regular pentagon.^[13] Likewise, the cosine of π/12 (15°) arises from the half-angle formula applied to cos(π/6) = √3/2, yielding a nested radical that denests to \cos(\pi/12) = (\sqrt{6} + \sqrt{2})/4. A prominent derivation stems from the triple-angle formula \sin(3\theta) = 3\sin\theta - 4\sin^3\theta. To solve for \sin(\theta/3), substitute \phi = \theta/3, leading to the cubic equation $4x^3 - 3x + \sin\theta = 0 where x = \sin\phi. The real root of this depressed cubic can be expressed using nested square roots via the trigonometric solution method, which leverages the identity's connection to angle trisection and yields explicit radical forms for certain θ.^[14] For more intricate angles, such as sin(π/60) (3°), the exact value requires successive applications of half-angle and triple-angle formulas, producing a deeply nested radical expression involving multiple levels of square roots; this can be simplified through denesting techniques to a form with sums of radicals.^[15]

In Solving Cubic Equations

Cardano's formula provides an explicit solution to the general cubic equation ax^3 + bx^2 + cx + d = 0 in terms of radicals, as published in Gerolamo Cardano's 1545 work Ars Magna. The formula first reduces the equation to the depressed form x^3 + px + q = 0 via the substitution x = y - \frac{b}{3a}, eliminating the quadratic term.^[16] The solutions then involve cube roots, with square roots appearing in the discriminants inside those cube roots, forming nested radicals. Specifically, one root is given by

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} + \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} - \sqrt{\left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3}}.

The other roots follow from multiplying by cube roots of unity.^[17] This structure highlights the nested nature, where the inner square root determines the branch and reality of the expression. The term under the square root, \Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3, acts as a discriminant indicator. When \Delta > 0, there is one real root and two complex conjugate roots, with the expression using real radicals. When \Delta = 0, multiple roots occur. However, in the casus irreducibilis (\Delta < 0), all three roots are real, but the formula requires cube roots of complex numbers, whose imaginary parts cancel in the sum to yield real values. This case necessitates nested radicals and cannot be simplified to non-nested real radical expressions without complex intermediates, as proven by results tied to Abel's theorem on the unsolvability of higher-degree equations by radicals in general. Alternatively, the real roots can be expressed using trigonometric functions. A classic example is the depressed cubic x^3 - 3x - 1 = 0, where p = -3, q = -1, and \Delta = -\frac{3}{4} < 0, placing it in the casus irreducibilis with three real roots. Applying Cardano's formula yields

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{1}{2} + \sqrt{-\frac{3}{4}}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{1}{2} - \sqrt{-\frac{3}{4}}},

where the inner square roots are imaginary, but the principal real root emerges as approximately 1.879 after cancellation. The full set of roots requires the other cube root branches. In such irreducible cases, the trigonometric form provides a real-valued alternative without complexes.

Infinite Nested Radicals

Nested Square Roots of 2

The infinite nested square root of 2 is defined as the limit x = \lim_{n \to \infty} x_n, where the sequence is given by x_1 = \sqrt{2} and x_{n+1} = \sqrt{2 + x_n} for n \geq 1.^[18] Assuming the limit exists, it satisfies the equation x = \sqrt{2 + x}. Squaring both sides yields x^2 = 2 + x, or x^2 - x - 2 = 0, which factors as (x - 2)(x + 1) = 0. The positive solution is x = 2, as the expression involves nonnegative terms.^[18] To verify uniqueness in the positive reals, note that the function f(x) = \sqrt{2 + x} is increasing and concave down for x \geq 0, with the fixed point at x = 2 being attractive since f'(2) = 1/(2\sqrt{3}) < 1.^[19] The sequence \{x_n\} is increasing and bounded above by 2, hence converges by the monotone convergence theorem.^[18] Specifically, x_1 = \sqrt{2} \approx 1.414 < 2, and by induction, if x_n < 2, then x_{n+1} = \sqrt{2 + x_n} < \sqrt{2 + 2} = 2; moreover, x_{n+1} > x_n since x_n^2 < 2 + x_n follows from x_n < 2. For the general case of constant coefficients, Herschfeld's convergence theorem confirms that the nested radical \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} converges because the sequence $2^{1/2^n} is bounded (it approaches 1 as n \to \infty).^[18] The finite approximations admit a trigonometric closed form that highlights a geometric interpretation related to angle bisection in polygonal constructions. The n-fold nested radical x_n = \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2}}} (with n square roots) equals $2 \cos\left( \frac{\pi}{2^{n+1}} \right).^[20] This follows from the half-angle formula $2 \cos(\theta/2) = \sqrt{2 + 2 \cos \theta} for $0 < \theta < \pi/2, starting with x_1 = \sqrt{2} = 2 \cos(\pi/4) and iterating to halve the angle successively.^[20] As n \to \infty, \frac{\pi}{2^{n+1}} \to 0, so x_n \to 2 \cos(0) = 2, linking the algebraic limit to the geometry of dyadic angle divisions, as in approximations to regular polygons with $2^{n+1} sides.^[21]

Ramanujan's Infinite Radicals

In 1911, Srinivasa Ramanujan posed a problem involving an infinite nested radical in the Journal of the Indian Mathematical Society, marking one of his early published contributions that showcased his intuitive grasp of infinite expressions. This work, detailed in Question 289 on page 90 of volume 3, connected nested radicals to broader themes in his notebooks, including analogies with continued fractions and functional equations.^[22] The solution appeared in volume 4 of the journal in 1912, affirming the radical's value through a clever algebraic verification. The specific identity is given by the infinite nested radical

L = \sqrt{1 + 2\sqrt{1 + 3\sqrt{1 + 4\sqrt{1 + \cdots}}}}

which equals 3.^[22] Finite truncations of this expression, starting from the innermost term \sqrt{1} = 1 and building outward, yield approximations that increase monotonically and approach 3 from below. For example, the truncation at n=3 gives \sqrt{1 + 2\sqrt{1 + 3\sqrt{1}}} \approx 2.236, while at n=10 it reaches approximately 2.875, demonstrating steady convergence to the limit value.^[10] A general form emerges by considering the nested radical starting at an arbitrary positive integer coefficient k \geq 1:

L_k = \sqrt{1 + k \sqrt{1 + (k+1)\sqrt{1 + (k+2)\sqrt{1 + \cdots}}}}.

This equals k + 1.^[10] For k=1, it recovers a related expression aligning with the original via index shift, yielding 2, but the canonical case starts at k=2 to give 3. The pattern holds for successive starting points, such as L_3 = 4 and L_4 = 5, producing integer values consistent with Ramanujan's interest in exact evaluations.^[22] To verify the identity, assume the expression converges to L. Then L^2 = 1 + 2M, where M = \sqrt{1 + 3N} is the inner nested radical starting at coefficient 3, and similarly M^2 = 1 + 3N, with N^2 = 1 + 4P, and so on. Setting L = 3 gives $9 = 1 + 2M, so M = 4; then $16 = 1 + 3N, so N = 5; continuing yields $25 = 1 + 4P, so P = 6, and the pattern Q = 7, etc., persists indefinitely without contradiction, as the infinite nesting has no terminating term. This recursive relation L_k^2 = 1 + k L_{k+1} with L_k = k + 1 holds algebraically:

(k + 1)^2 = k^2 + 2k + 1 = 1 + k(k + 2),

confirming consistency across the chain.^[10] Ramanujan's approach highlights the nested radical's ties to integer arithmetic and infinite processes, influencing later generalizations in analytic number theory.^[22]

Viète's Expression for π

In 1593, French mathematician François Viète published the first known infinite product expression for π, derived through geometric considerations involving inscribed polygons and trigonometric identities, predating infinite series expansions by nearly a century.^[23] This formula represents a pioneering use of infinite nested radicals in European mathematics, marking an early step toward modern analysis by expressing π as the limit of a recursive process.^[23] Viète's formula is given by the infinite product

\frac{2}{\pi} = \sqrt{\frac{1}{2}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}} \times \sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2} + \frac{1}{2}\sqrt{\frac{1}{2}}}} \times \cdots

or, more compactly, as the limit of the partial products p_n = \prod_{k=1}^n a_k, where a_1 = \sqrt{1/2} and a_k = \sqrt{1/2 + a_{k-1}/2} for k \geq 2, converging to $2/\pi.^[24] Equivalently, it can be expressed as \frac{2}{\pi} = \prod_{k=1}^\infty \cos\left(\frac{\pi}{2^{k+1}}\right).^[24] The derivation stems from the infinite product representation of the sinc function, \operatorname{sinc}(x) = \prod_{k=1}^\infty \cos\left(\frac{x}{2^k}\right), discovered by Viète.^[24] Evaluating at x = \pi/2 yields \operatorname{sinc}(\pi/2) = 2/\pi. To obtain the nested radical form, apply the half-angle formula iteratively: \cos(\theta/2) = \sqrt{(1 + \cos \theta)/2}, starting from \cos(\pi/4) = \sqrt{1/2} and halving the angle repeatedly, which generates the recursive structure of the radicals.^[24] Finite truncations of the product provide successive approximations to π with linear convergence, where the error decreases proportionally to the number of terms. For example, the first term gives approximately 2.83, the first two terms yield about 3.06, the first three terms approximate 3.12, and the first four terms give roughly 3.14.^[1] Viète himself used this method with a 393,216-sided polygon (equivalent to 16 iterations) to compute π to 10 decimal places.^[23]

Nested Cube Roots

Infinite nested cube roots take the form

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{a + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{a + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{a + \cdots}}}

, where the expression converges to a value satisfying the equation

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{a + x}

. Cubing both sides gives x^3 = a + x, or the depressed cubic equation

x^3 - x - a = 0.

^[25] This cubic equation has discriminant D = 4 - 27a^2. The nature of the roots depends on the value of a: if |a| < \frac{2}{3\sqrt{3}} \approx 0.384, then D > 0 and there are three distinct real roots; if |a| = 0.384, then D = 0 and there are two real roots (one repeated); if |a| > 0.384, then D < 0 and there is one real root and two complex conjugate roots.^[26] The solutions can be found using Cardano's formula. For the general depressed cubic x^3 + px + q = 0, the real root (when D < 0) is given by

x = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} + \sqrt{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{-\frac{q}{2} - \sqrt{\left( \frac{q}{2} \right)^2 + \left( \frac{p}{3} \right)^3}},

where here p = -1 and q = -a. When D > 0, the three real roots can be expressed using the trigonometric form:

x_k = 2 \sqrt{-\frac{p}{3}} \cos\left( \frac{1}{3} \arccos\left( \frac{-q}{2} \left( -\frac{p}{3} \right)^{-\frac{3}{2}} \right) - \frac{2\pi k}{3} \right), \quad k = 0,1,2.

For p = -1, this simplifies to

x_k = \frac{2}{\sqrt{3}} \cos\left( \frac{1}{3} \arccos\left( \frac{3\sqrt{3} a}{2} \right) - \frac{2\pi k}{3} \right),

with k = 0 yielding the largest real root.^[26] Unlike the quadratic case for nested square roots where there is always a unique positive real root for a \geq 0, the cubic equation here has three distinct real roots only when |a| < \frac{2}{3\sqrt{3}} \approx 0.384; nevertheless, for a \geq 0, the nested radical converges to the largest (positive) real root, as the defining sequence x_0 = 0,

x_{n+1} = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{a + x_n}

is increasing and bounded above by this root.^[25] A representative example is a = 1, where the equation is x^3 - x - 1 = 0. This has one real root, known as the plastic constant \rho, with approximate value 1.3247. Using Cardano's formula,

\rho = \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{1}{2} + \sqrt{\frac{23}{108}}} + \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=3&&&citation_type=wikipedia}}{\frac{1}{2} - \sqrt{\frac{23}{108}}}.

The sequence converges to \rho, illustrating the primary algebraic solution for such nested cube roots.^[27]

Herschfeld's Convergence Theorem

Herschfeld's convergence theorem, established in 1935, provides a necessary and sufficient condition for the convergence of infinite nested radicals of the form x = \sqrt{a_1 + \sqrt{a_2 + \sqrt{a_3 + \cdots}}}, where (a_n)_{n=1}^\infty is a sequence of nonnegative real numbers. The theorem states that the expression converges to a finite limit if and only if \sup_{n \geq 1} a_n^{1/2^n} < \infty. To prove the "if" direction, assume M = \sup_{n \geq 1} a_n^{1/2^n} < \infty. Define the partial nested radicals recursively backward: let x_{n+1}^{(n)} = 0 (or more precisely, truncate at some point, but for bounding, consider the increasing sequence of finite nests). The forward partial approximations x_k^{(n)} for k = 1, \dots, n, with x_n^{(n)} = \sqrt{a_n} and x_k^{(n)} = \sqrt{a_k + x_{k+1}^{(n)}} for k < n, form an increasing sequence bounded above. Using induction on the nesting level, one can show that x_1^{(n)} is bounded by a value involving M and the golden ratio \phi = (1 + \sqrt{5})/2 \approx 1.618. Specifically, the error bound satisfies |x_n - L| \leq r^{2^n}, where L is the limit and r = 1/\phi < 1 is the contraction factor, derived from the geometric series arising in the inductive step via the concavity of the square root function. This ensures the sequence x_1^{(n)} converges as n \to \infty.^[28] For the "only if" direction, assume the nested radical converges to a finite limit L \geq 0. The partial approximations x_k^{(n)} converge uniformly to values bounded by L, and working backward recursively, a_n = (x_n^{(n)})^2 - x_{n+1}^{(n+1)} + \epsilon_n, where \epsilon_n \to 0. Since $0 \leq x_k^{(n)} \leq L for all k, n, it follows that a_n \leq (L + \delta)^{2^n} for any \delta > 0 and sufficiently large n, implying a_n^{1/2^n} \leq L + \delta. Thus, \sup_n a_n^{1/2^n} \leq L < \infty. This backward recursion leverages the assumed convergence to derive the boundedness condition.^[28] This theorem has significant applications in determining the convergence of specific infinite nested radicals. For instance, when a_n = 2 for all n, a_n^{1/2^n} = 2^{1/2^n} \to 1 < \infty, so the expression \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} converges to 2. In contrast, if a_n = 2^{n \cdot 2^n}, then a_n^{1/2^n} = 2^n, which is unbounded, and the nested radical diverges to infinity. These examples illustrate how the theorem distinguishes convergent cases, such as constant sequences, from divergent ones requiring super-exponential growth in a_n.

Recent Developments

Generalizations of Ramanujan's Nested Radicals

In recent mathematical literature, significant generalizations of Ramanujan's nested radicals have extended the original finite and infinite forms to encompass all positive integers n, allowing the expressions to evaluate to arbitrary integers beyond the classical case of 3. A 2025 study introduces a systematic approach using the difference-of-squares identity p^2 - q^2 = (p - q)(p + q) to construct convergent infinite nested radicals that equal any positive integer x. This work defines a "Radiciatory" operator R(x) = \sqrt{x + R(x + d)}, where d is a nonzero parameter, enabling representations such as \sqrt{x + \sqrt{x + d + \sqrt{x + 2d + \cdots}}} = x. Numerical validations and tables confirm convergence for various x and integer d, broadening Ramanujan's insight into algebraic symmetries for integer-valued limits.^[29] Key extensions incorporate variable coefficients b_k in forms like \sqrt{a_1 + b_1 \sqrt{a_2 + b_2 \sqrt{a_3 + \cdots}}}, where adjustments to b_k allow convergence to non-integers while maintaining stability under Herschfeld's theorem conditions. For instance, selecting non-integer d in the recursive structure yields limits like rational or irrational values, demonstrated through bounding sequences that ensure monotonic increase and upper bounds. These generalizations preserve the nested structure but introduce flexibility in parameters to target specific non-integer outcomes, such as fractions or algebraic numbers.^[29] Methods for deriving these identities rely on recursive definitions and induction proofs. Consider the example for n=4: the nested radical \sqrt{4 + \sqrt{9 + \sqrt{16 + \sqrt{25 + \cdots}}}} = 4, where the inner terms follow a pattern adjusted by d=5, yielding \sqrt{4 + \sqrt{4 + 5 + \sqrt{4 + 10 + \sqrt{4 + 17 + \cdots}}}} = 4. Proof proceeds by induction: assume the form holds for the tail starting at k, then verify R(4) = \sqrt{4 + R(9)} = 4 using the algebraic identity, with base cases confirmed numerically and convergence established via increasing bounded sequences. Similar recursive techniques apply to higher n.^[29] A parallel 2024 generalization elevates Ramanujan's square-root case to arbitrary root orders n \geq 2, using binomial expansions in functional equations. The core theorem states x + 1 = \sqrt{1 + \binom{n}{1} x + \cdots + \binom{n}{n-1} x^{n-1} + x^n \sqrt{ \cdots }}, where the nested form converges to the integer x+1 for suitable starting x. For n=4, setting x=3 yields 4 via

\sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{1 + 4 \cdot 3 + 6 \cdot 3^2 + 3^3 \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{1 + 4(3+3) + 6(3+3)^2 + (3+3)^3 \sqrt{{grok:render&&&type=render_inline_citation&&&citation_id=4&&&citation_type=wikipedia}}{\cdots}}} = 4

, proved by cubing (or higher powering) both sides and matching coefficients recursively, with convergence shown through monotonicity and bounds. This approach, while focused on integer limits, hints at extensions to non-integers via parameter scaling. A variant identity appears in \sqrt{1 + \sqrt{4 + \sqrt{9 + \cdots + \sqrt{n^2 + \cdots}}}} = \frac{n+1}{2}, derived similarly for specific finite truncations generalizing to infinite cases with adjusted coefficients.^[22]

New Iterative Formulas for π

In 2025, researchers Sanjar M. Abrarov, Rehan Siddiqui, Rajinder Kumar Jagpal, and Brendan M. Quine introduced a novel iterative method for computing π that leverages nested radicals with roots of 2, offering improved efficiency for high-precision calculations.^[30] This approach builds on historical nested radical expressions, such as Viète's 16th-century formula, but incorporates modern iterative schemes derived from arctangent identities to achieve faster convergence.^[30] The method generates two-term Machin-like formulas, where π is approximated through combinations of arctangents, with nested radicals providing the key building blocks for the arguments. The core of the method involves defining a sequence of nested radicals given by

c_n = \sqrt{2 + c_{n-1}}, \quad c_0 = 0,

which converges to values related to trigonometric functions, specifically enabling approximations like c_n / \sqrt{2 - c_{n-1}}.^[30] These terms are then used to initialize an iterative process for the variables v_n:

v_n = \frac{1}{2} \left( v_{n-1} - \frac{1}{v_{n-1}} \right), \quad n = 2, 3, \dots, k,

starting from v_1 = \gamma_k, a value derived from the nested radicals.^[30] This iteration yields the Machin-like formula

\frac{\pi}{4} = 2^{k-1} \arctan\left( \frac{1}{v_1} \right) + \arctan\left( \frac{v_k - 1}{v_k + 1} \right),

allowing for systematic computation of π digits by increasing k.^[30] The nested radicals ensure that the initial v_1 captures the geometric structure of the approximation, facilitating numerical evaluation in software like Mathematica. The method exhibits cubic convergence in a related iterative refinement step, where the error satisfies \varepsilon_{n+1} = \varepsilon_n^3 / 6 + O(\varepsilon_n^5), making it suitable for arbitrary-precision arithmetic.^[30] Compared to Viète's infinite product, which relies on successive multiplications of nested square roots, this iterative approach reduces computational steps by transforming the problem into a finite arctangent sum with quadratic growth in precision per iteration.^[30] It also outperforms Archimedes' classical polygonal method, which requires exponentially more iterations for equivalent precision (e.g., over 100 iterations for 10 digits versus fewer than 10 here), due to its algebraic nature over geometric bounds.^[30] Numerical demonstrations highlight its efficiency: for k=50, the formula computes over 500,000 digits of π, while at k=3000, a single-term variant yields 902 digits and the two-term version doubles that to 1,805 digits, approximately five times faster than comparable arctangent-based predecessors.^[30] This makes the method particularly advantageous for applications requiring rapid high-precision π evaluation, such as in scientific simulations and cryptographic algorithms.

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### Summary of Ramanujan's Original Infinite Nested Radical
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