Lebl-Basic-Analysis-I · Problem 2.2.10

Sequences and Series

Mathematics · facts-about-limits-of-sequences

Problem

(Challenging) Let

\{x_n\}_{n=1}^\infty

be a convergent sequence such that

x_n \geq 0

and

k \in \mathbb{N}

. Then

\lim_{n\to\infty} x_n^{1/k} = \left(\lim_{n\to\infty} x_n\right)^{1/k}.

Hint: Find an expression

q

such that

\dfrac{x_n^{1/k} - x^{1/k}}{x_n - x} = \dfrac{1}{q}