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.
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