Question

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

score 302 · Answer 1 · 2018-12-14 06:12:31

302

Views

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

Published on 14 Dec 2018 - 6:12

score 1.1k · Answer 2 · 2018-12-14 14:20:56

1.1k

Views

How to show that $\sin(n)$ does not converge ONLY by using Cauchy's criterion?

Published on 14 Dec 2018 - 14:20

score 349 · Answer 3 · 2018-12-16 15:39:28

349

Views

Show that if two metrics induce the same topology, one metric space is compact iff the other one is.

Published on 16 Dec 2018 - 15:39

score 394 · Answer 4 · 2018-12-17 13:50:37

394

Views

Proving that every Cauchy sequence in measure converges in measure

Published on 17 Dec 2018 - 13:50

score 78 · Answer 5 · 2018-12-18 11:29:07

78

Views

Identical Cauchy sequences and continuity.

Published on 18 Dec 2018 - 11:29

score 110 · Answer 6 · 2018-12-18 13:57:22

110

Views

Proof verification that $\{x_n\} = 0,\underbrace{77\dots 7}_{\text{n times}}$ is a Cauchy sequence.

Published on 18 Dec 2018 - 13:57

score 42 · Answer 7 · 2018-12-19 14:17:16

42

Views

Proof verification of $x_n = \sum_{k=1}^n a_kq^k$ is Cauchy given $|a_k| \le C, |q| < 1, k\in\Bbb N$

Published on 19 Dec 2018 - 14:17

score 165 · Answer 8 · 2026-02-22 23:11:39

165

Views

Is the notion of Cauchy sequences definable in a bornological topological space?

Published on 22 Feb 2026 - 23:11

score 323 · Answer 9 · 2018-12-19 18:42:57

323

Views

Prove that $\{x_n\}$ is a Cauchy sequence iff $\forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$

Published on 19 Dec 2018 - 18:42

score 79 · Answer 10 · 2018-12-20 15:15:30

79

Views

Proof verification of $x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$ diverges by the negation of Cauchy

Published on 20 Dec 2018 - 15:15

score 142 · Answer 11 · 2018-12-21 16:56:38

142

Views

Sequence of function on $\mathbb{R}$ Cauchy iff convergent

Published on 21 Dec 2018 - 16:56

score 131 · Answer 12 · 2018-12-24 17:06:12

131

Views

Proof verification. $\{x_n\}$ is a sequence such that $|x_{n+1} - x_n| \le C\alpha^n$ for $\alpha\in (0, 1), n\in\Bbb N$. Prove $x_n$ converges.

Published on 24 Dec 2018 - 17:06

score 203 · Answer 13 · 2018-12-26 11:22:14

203

Views

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Published on 26 Dec 2018 - 11:22

score 329 · Answer 14 · 2018-12-28 15:47:19

Please help me to understand a proof that $a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$ is a Cauchy sequence

score 68 · Answer 15 · 2018-12-28 19:34:26

68

Views

Meaning of being a Cauchy sequence

Published on 28 Dec 2018 - 19:34

List Question

Characterization of the sequence $x_1=\cos(x), x_{n+1}=\cos(x_n)$ where $x>0.$

How to show that $\sin(n)$ does not converge ONLY by using Cauchy's criterion?

Show that if two metrics induce the same topology, one metric space is compact iff the other one is.

Proving that every Cauchy sequence in measure converges in measure

Identical Cauchy sequences and continuity.

Proof verification that $\{x_n\} = 0,\underbrace{77\dots 7}_{\text{n times}}$ is a Cauchy sequence.

Proof verification of $x_n = \sum_{k=1}^n a_kq^k$ is Cauchy given $|a_k| \le C, |q| < 1, k\in\Bbb N$

Is the notion of Cauchy sequences definable in a bornological topological space?

Prove that $\{x_n\}$ is a Cauchy sequence iff $\forall \epsilon > 0\ \exists N: \forall n > N \implies |x_n - x_N| < \epsilon$

Proof verification of $x_n = {1\over2^2}+{2\over3^2}+\cdots+{n\over(n+1)^2}$ diverges by the negation of Cauchy

Sequence of function on $\mathbb{R}$ Cauchy iff convergent

Proof verification. $\{x_n\}$ is a sequence such that $|x_{n+1} - x_n| \le C\alpha^n$ for $\alpha\in (0, 1), n\in\Bbb N$. Prove $x_n$ converges.

Show that $S_n=1+{x\over1!}+{x^2\over2!}+\cdots+{x^n\over n!}$ converges for $n\in\Bbb N,\ x \in\Bbb R$ without using Taylor series.

Please help me to understand a proof that $a_n = \frac{\tan 1}{2} + \frac{\tan2 }{2^2} + \dots + \frac{\tan n}{2^n}$ is a Cauchy sequence

Meaning of being a Cauchy sequence

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