continued square root | iterated radical | nested root | radical denesting

Expressions of the form lim_(k->∞) x_0 + sqrt(x_1 + sqrt(x_2 + sqrt(... + x_k))) are called nested radicals. Herschfeld proved that a nested radical of real nonnegative terms converges iff (x_n)^(2^(-n)) is bounded. He also extended this result to arbitrary powers (which include continued square roots and continued fractions as well), a result is known as Herschfeld's convergence theorem.

Bolyai expansion | continued fraction | golden ratio | Herschfeld's convergence theorem | nested radical constant | Paris constant | pi formulas | power tower | Ramanujan log-trigonometric integrals | silver constant | Somos's quadratic recurrence constant | square root