Question

Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous

score 53 · Answer 1 · 2020-06-17 12:10:16

53

Views

Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous

Published on 17 Jun 2020 - 12:10

score 227 · Answer 2 · 2020-06-18 16:05:25

227

Views

Why Folland's Advanced Calculus is so strict about uniform convergence?

Published on 18 Jun 2020 - 16:05

score 157 · Answer 3 · 2020-06-18 19:20:19

157

Views

Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.

Published on 18 Jun 2020 - 19:20

score 85 · Answer 4 · 2020-06-19 12:04:09

85

Views

Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

Published on 19 Jun 2020 - 12:04

score 43 · Answer 5 · 2020-06-19 19:17:43

43

Views

Show that $f(x):=\sum\limits_{n=0}^{\infty}\frac{1}{n}h(2^{n}x),$ where $h$ is a piecewise function, converges uniformly on $[0,1]$

Published on 19 Jun 2020 - 19:17

score 360 · Answer 6 · 2026-04-03 10:53:44

360

Views

Converges uniformly on an arbitrary closed disc implies on every compact subsets

Published on 03 Apr 2026 - 10:53

score 105 · Answer 7 · 2026-03-29 10:08:55

105

Views

Convergence in measure of $\cos(\frac{x}{n})$ for $x\in[0,\pi]$.

Published on 29 Mar 2026 - 10:08

score 168 · Answer 8 · 2020-06-22 17:09:00

168

Views

Let $f_n(x):\mathbb R \to \mathbb R$ be defined by $f_n(x)=\frac{x}{1+nx^2}$. Which of the following statements are correct?

Published on 22 Jun 2020 - 17:09

score 39 · Answer 9 · 2026-04-01 16:11:02

39

Views

Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$

Published on 01 Apr 2026 - 16:11

score 125 · Answer 10 · 2026-02-24 19:37:51

125

Views

Show that $\phi(x):=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}(1+\frac{x^{2}}{n})^{n}}$ is differentiable on $\mathbb{R}$.

Published on 24 Feb 2026 - 19:37

score 1k · Answer 11 · 2020-06-25 10:51:32

1k

Views

Uniform Convergence of the Complex Exponential Function

Published on 25 Jun 2020 - 10:51

score 68 · Answer 12 · 2020-06-25 11:11:15

68

Views

How do I prove that a sequence of functions is convergent?

Published on 25 Jun 2020 - 11:11

score 27 · Answer 13 · 2020-06-25 15:51:06

27

Views

I need to determine the domain of convergence and whether the series converge uniformly.

Published on 25 Jun 2020 - 15:51

score 93 · Answer 14 · 2020-06-26 03:42:35

93

Views

About uniform convergence.

Published on 26 Jun 2020 - 3:42

score 92 · Answer 15 · 2026-02-24 19:37:20

92

Views

Check the pointwise convergence and uniform convergence of a sequence of functions.

Published on 24 Feb 2026 - 19:37

List Question

Show that the sum function $f(x) = \sum_{n=1}^\infty \frac{1}{ \sqrt{n} } (exp(-x^2/n)-1)$ is continous

Why Folland's Advanced Calculus is so strict about uniform convergence?

Show that $\sum_{n=1}^{\infty}\frac{x^{2}}{x^{2}+n^{2}}$ does not converge uniformly on $(-\infty,\infty)$.

Show $ f_n\xrightarrow{L^1}f\iff f_n\to f \text{ in measure} \iff f_n\to f \text{ almost uniformly} \iff f_n\to f \text{ a.e.}$

Show that $f(x):=\sum\limits_{n=0}^{\infty}\frac{1}{n}h(2^{n}x),$ where $h$ is a piecewise function, converges uniformly on $[0,1]$

Converges uniformly on an arbitrary closed disc implies on every compact subsets

Convergence in measure of $\cos(\frac{x}{n})$ for $x\in[0,\pi]$.

Let $f_n(x):\mathbb R \to \mathbb R$ be defined by $f_n(x)=\frac{x}{1+nx^2}$. Which of the following statements are correct?

Show that $f(s)=\sum \frac{1}{n^s}$ is continuous for $Re(s)>1$

Show that $\phi(x):=\sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}(1+\frac{x^{2}}{n})^{n}}$ is differentiable on $\mathbb{R}$.

Uniform Convergence of the Complex Exponential Function

How do I prove that a sequence of functions is convergent?

I need to determine the domain of convergence and whether the series converge uniformly.

About uniform convergence.

Check the pointwise convergence and uniform convergence of a sequence of functions.

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