Continued Square Root

Expressions of the form

$$\lim_{k\to\infty} x_0+\sqrt{x_1+\sqrt{x_2+\sqrt{\ldots+x_k}}}.$$

Herschfeld (1935) proved that a continued square root of Real Nonnegative terms converges Iff $(x_n)^{2^{-n}}$ is bounded. He extended this result to arbitrary Powers (which include continued square roots and Continued Fractions as well), which is known as Herschfeld's Convergence Theorem.

See also Continued Fraction, Herschfeld's Convergence Theorem, Nested Radical, Square Root

References

Herschfeld, A. ``On Infinite Radicals.'' Amer. Math. Monthly 42, 419-429, 1935.

Pólya, G. and Szegö, G. Problems and Theorems in Analysis, Vol. 1. New York: Springer-Verlag, 1997.

Sizer, W. S. ``Continued Roots.'' Math. Mag. 59, 23-27, 1986.