Herschfeld's Convergence Theorem

For real, Nonnegative terms $x_n$ and Real $p$ with $0<p<1$, the expression

$$\lim_{k\to\infty} x_0+\left({x_1+\left({x_2+\left({\ldots+(x_k)^p}\right)^p}\right)^p}\right)^p$$

converges Iff $(x_n)^{p^n}$ is bounded.

See also Continued Square Root

References

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

Jones, D. J. ``Continued Powers and a Sufficient Condition for Their Convergence.'' Math. Mag. 68, 387-392, 1995.