Question

If $\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = a > 0$, prove that $\lim_{n \to \infty} \sqrt[n]{x_n} = a$

score 1.1k · Answer 1 · 2026-04-01 05:06:26

1.1k

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If $\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = a > 0$, prove that $\lim_{n \to \infty} \sqrt[n]{x_n} = a$

Published on 01 Apr 2026 - 5:06

score 165 · Answer 2 · 2026-03-30 16:41:12

165

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A Tauberian theorem for a quotient of power series, the limit on the boundary

Published on 30 Mar 2026 - 16:41

score 44 · Answer 3 · 2018-08-20 14:49:09

44

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Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$

Published on 20 Aug 2018 - 14:49

score 448 · Answer 4 · 2026-03-22 16:32:11

448

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Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

Published on 22 Mar 2026 - 16:32

score 186 · Answer 5 · 2018-08-21 08:48:07

186

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Convergence and limit of recursive sequence

Published on 21 Aug 2018 - 8:48

score 153 · Answer 6 · 2018-08-21 13:53:30

153

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Convergence of $\sum_{n=1}^\infty \frac{1}{n^{a_n}}$

Published on 21 Aug 2018 - 13:53

score 488 · Answer 7 · 2018-08-22 03:49:01

488

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Geometric mean of a positive convergent sequence

Published on 22 Aug 2018 - 3:49

score 154 · Answer 8 · 2018-08-22 13:21:12

154

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Why is $\sum_{k=0}^{\infty} {1/2 \choose k} = \sqrt{2}$?

Published on 22 Aug 2018 - 13:21

score 1.9k · Answer 9 · 2018-08-22 14:47:48

1.9k

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What can we say about $a_n$ if $\sum a_n/n$ converges?

Published on 22 Aug 2018 - 14:47

score 33 · Answer 10 · 2018-08-22 17:06:16

33

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Extending convergence from a point to an interval

Published on 22 Aug 2018 - 17:06

score 62 · Answer 11 · 2018-08-23 08:54:33

62

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Convergence of Laurent series principal part

Published on 23 Aug 2018 - 8:54

score 189 · Answer 12 · 2026-03-27 18:27:44

189

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Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $

Published on 27 Mar 2026 - 18:27

score 115 · Answer 13 · 2018-08-23 17:23:49

115

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Proving convergence almost surely

Published on 23 Aug 2018 - 17:23

score 5.1k · Answer 14 · 2026-02-23 06:38:02

5.1k

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Summation problem: $f(x)=1+\sum_{n=1}^{\infty}\frac{x^n}{n}$

Published on 23 Feb 2026 - 6:38

score 1.1k · Answer 15 · 2018-08-23 20:33:14

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Radius of convergence of $\sum_{k=1}^{\infty} {{k^{\sqrt k}}x^{k}}$

Published on 23 Aug 2018 - 20:33

List Question

If $\lim_{n \to \infty} \frac{x_{n+1}}{x_n} = a > 0$, prove that $\lim_{n \to \infty} \sqrt[n]{x_n} = a$

A Tauberian theorem for a quotient of power series, the limit on the boundary

Convergence of $\sum_{n=1}^{\infty}\left(\frac{n}{n^2+1}\right)^{k(n)}\,\,\,\,\,;\,\,k(n)=\frac{1}{\cos\left(\frac{1}{\ln^{a}(n)}\right)}$

Pointwise convergence of sequence $(f_n)_n$ of functions to $f$ and changing limits

Convergence and limit of recursive sequence

Convergence of $\sum_{n=1}^\infty \frac{1}{n^{a_n}}$

Geometric mean of a positive convergent sequence

Why is $\sum_{k=0}^{\infty} {1/2 \choose k} = \sqrt{2}$?

What can we say about $a_n$ if $\sum a_n/n$ converges?

Extending convergence from a point to an interval

Convergence of Laurent series principal part

Nature of infinite series $ \sum\limits_{n\geq 1}\left[\frac{1}{n} - \log(1 + \frac{1}{n})\right] $

Proving convergence almost surely

Summation problem: $f(x)=1+\sum_{n=1}^{\infty}\frac{x^n}{n}$

Radius of convergence of $\sum_{k=1}^{\infty} {{k^{\sqrt k}}x^{k}}$

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