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Herschfeld's Convergence Theorem

Herschfeld's convergence theorem states that for real, nonnegative terms x_n and real p with 0<p<1, the expression

lim_(k->infty)x_0+(x_1+(x_2+(...+(x_k)^p)^p)^p)^p

converges iff (x_n)^(p^n) is bounded.

See also

Nested Radical

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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.

Referenced on Wolfram|Alpha

Herschfeld's Convergence Theorem

Cite this as:

Weisstein, Eric W. "Herschfeld's Convergence Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HerschfeldsConvergenceTheorem.html

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