Question

Almost sure and order convergence are equivalent in $L_p$ spaces

score 92 · Answer 1 · 2026-04-07 16:39:45

92

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Almost sure and order convergence are equivalent in $L_p$ spaces

Published on 07 Apr 2026 - 16:39

score 1.6k · Answer 2 · 2018-12-14 02:20:29

1.6k

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Finding $p$ such that $\sum\limits_{n=1}^\infty n(1+n^2)^p$ converges. Check my work.

Published on 14 Dec 2018 - 2:20

score 479 · Answer 3 · 2026-03-27 01:00:06

479

Views

Given $f(x)$ is integrable on $[0, 1]$ and $0 < f(x) < 1$, prove that $\int_{0}^{1} (f(x))^{n} \mathop{dx}$ converges to $0$.

Published on 27 Mar 2026 - 1:00

score 50 · Answer 4 · 2018-12-14 02:27:25

50

Views

When the series $\sum_{n=2}^{+\infty }(\sqrt[n]{n}-1)^a$ is convergent depending on $a\in\Bbb R$?

Published on 14 Dec 2018 - 2:27

score 303 · Answer 5 · 2026-03-26 02:56:07

303

Views

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

Published on 26 Mar 2026 - 2:56

score 1.1k · Answer 6 · 2026-03-26 04:32:01

1.1k

Views

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

Published on 26 Mar 2026 - 4:32

score 37 · Answer 7 · 2026-02-23 10:22:05

37

Views

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

Published on 23 Feb 2026 - 10:22

score 222 · Answer 8 · 2018-12-14 16:40:13

222

Views

$\sum a_n b_n$ when $\sum a_n$ convergent and $\{b_n\}$ nonnegative

Published on 14 Dec 2018 - 16:40

score 492 · Answer 9 · 2018-12-14 18:32:50

492

Views

Any two of the subsequences $\{a_{2k}\}, \{a_{2k+1}\}, \{a_{3k}\}$ converge do not imply $\{a_k\}$ converges?

Published on 14 Dec 2018 - 18:32

score 48 · Answer 10 · 2026-03-30 01:47:03

48

Views

What is the name of this series test?

Published on 30 Mar 2026 - 1:47

score 332 · Answer 11 · 2026-03-29 15:40:30

332

Views

Is $\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }$ uniformly convergent on $(-\pi , \pi)$?

Published on 29 Mar 2026 - 15:40

score 40 · Answer 12 · 2026-03-30 10:37:43

40

Views

Gaussian variable on $l^2$ (Exercise 3.5 from Hairer's lecture notes).

Published on 30 Mar 2026 - 10:37

score 147 · Answer 13 · 2026-03-27 04:57:08

147

Views

Determine whether $\sum_{n=1}^{\infty}\frac{n2^n}{n+3^n}$ converges.

Published on 27 Mar 2026 - 4:57

score 70 · Answer 14 · 2026-03-28 07:59:29

70

Views

Let $f_n:[1,\infty) \rightarrow \mathbb{R}$ be defined by $f_n(x)=\frac{1}{x}\chi_{[n,\infty)}(x)$. Does $\int f_n \to \int f$?

Published on 28 Mar 2026 - 7:59

score 104 · Answer 15 · 2018-12-15 01:12:49

104

Views

Prove that this series of functions converges on the interval [1,2].

Published on 15 Dec 2018 - 1:12

List Question

Almost sure and order convergence are equivalent in $L_p$ spaces

Finding $p$ such that $\sum\limits_{n=1}^\infty n(1+n^2)^p$ converges. Check my work.

Given $f(x)$ is integrable on $[0, 1]$ and $0 < f(x) < 1$, prove that $\int_{0}^{1} (f(x))^{n} \mathop{dx}$ converges to $0$.

When the series $\sum_{n=2}^{+\infty }(\sqrt[n]{n}-1)^a$ is convergent depending on $a\in\Bbb R$?

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?

Is $f_{n}$ is analytic on $(a, b)$ and $f_{n} \rightarrow f$ uniformly on $(a, b)$ then is $f$ analytic on $(a, b)$?

$\sum a_n b_n$ when $\sum a_n$ convergent and $\{b_n\}$ nonnegative

Any two of the subsequences $\{a_{2k}\}, \{a_{2k+1}\}, \{a_{3k}\}$ converge do not imply $\{a_k\}$ converges?

What is the name of this series test?

Is $\sum _{n=1} ^ \infty \frac{1}{(x+\pi)^2 \cdot n^2 }$ uniformly convergent on $(-\pi , \pi)$?

Gaussian variable on $l^2$ (Exercise 3.5 from Hairer's lecture notes).

Determine whether $\sum_{n=1}^{\infty}\frac{n2^n}{n+3^n}$ converges.

Let $f_n:[1,\infty) \rightarrow \mathbb{R}$ be defined by $f_n(x)=\frac{1}{x}\chi_{[n,\infty)}(x)$. Does $\int f_n \to \int f$?

Prove that this series of functions converges on the interval [1,2].

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