Question

Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

score 307 · Answer 1 · 2015-07-09 01:58:42

307

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Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

Published on 09 Jul 2015 - 1:58

score 100 · Answer 2 · 2015-07-09 10:42:52

100

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$f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

Published on 09 Jul 2015 - 10:42

score 58 · Answer 3 · 2026-04-02 06:19:27

58

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Uniform Convergence of $\sum_{n=0}^\infty (1-x^2)^2x^n$ on $[0,1]$; subsequent integral

Published on 02 Apr 2026 - 6:19

score 105 · Answer 4 · 2015-07-09 16:08:49

105

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$f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly .

Published on 09 Jul 2015 - 16:08

score 475 · Answer 5 · 2026-04-12 08:41:19

475

Views

Apply Stone- Weierstrass Theorem

Published on 12 Apr 2026 - 8:41

score 34 · Answer 6 · 2015-07-11 01:29:45

34

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The sequence $f_n '$ converge to a function $g$ on $[0,1]$ and $f'(x) = g(x), x \in (0,1), f'(1) \neq g'(1)$.

Published on 11 Jul 2015 - 1:29

score 96 · Answer 7 · 2026-04-12 10:22:45

96

Views

How to prove this limit of derivative

Published on 12 Apr 2026 - 10:22

score 202 · Answer 8 · 2015-07-12 04:20:12

202

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The sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$.

Published on 12 Jul 2015 - 4:20

score 84 · Answer 9 · 2015-07-15 07:48:35

84

Views

Uniform convergence of functions involving normal CDF

Published on 15 Jul 2015 - 7:48

score 187 · Answer 10 · 2015-07-18 01:41:24

187

Views

If $\sum a_n$ is a convergent series of real numbers then the following series are convergent.

Published on 18 Jul 2015 - 1:41

score 121 · Answer 11 · 2015-07-18 04:13:18

121

Views

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

Published on 18 Jul 2015 - 4:13

score 154 · Answer 12 · 2026-04-11 06:45:59

154

Views

measure theory problems and step functions

Published on 11 Apr 2026 - 6:45

score 150 · Answer 13 · 2026-04-11 09:51:58

150

Views

topology of uniform convergence on compacts and strong operator topology

Published on 11 Apr 2026 - 9:51

score 309 · Answer 14 · 2015-07-19 01:48:43

309

Views

the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$.

Published on 19 Jul 2015 - 1:48

score 296 · Answer 15 · 2015-07-19 12:31:01

296

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$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

Published on 19 Jul 2015 - 12:31

List Question

Uniform convergence and integration: $\lim_{n \to \infty} \int_{0}^{1} \frac{1}{1+x^2+\frac{x^4}{n}} dx=?$

$f_n ^{'}$ is uniformly convergent but the seq $f_n$ is not uniformly convergent.

Uniform Convergence of $\sum_{n=0}^\infty (1-x^2)^2x^n$ on $[0,1]$; subsequent integral

$f_n$ defined on $[-1,1]$ by $f_n(x) = |x|^{1+\frac1n}, \ x \in [-1,1]$ converges uniformly .

Apply Stone- Weierstrass Theorem

The sequence $f_n '$ converge to a function $g$ on $[0,1]$ and $f'(x) = g(x), x \in (0,1), f'(1) \neq g'(1)$.

How to prove this limit of derivative

The sequences $\{f_n\},\{f_n'\}$ both are uniformly convergent on $[0,1]$.

Uniform convergence of functions involving normal CDF

If $\sum a_n$ is a convergent series of real numbers then the following series are convergent.

$\int_a^b f(x)g(x)dx = \sum \int_a^b f_n(x)g(x)dx.$

measure theory problems and step functions

topology of uniform convergence on compacts and strong operator topology

the series $\sum_{1}^{\infty} \frac{\cos(nx)}{\{\log(n+1)\}^x}$ is uniformly convergent on any closed interval $[a,b]$ lying within $(0,2\pi)$.

$f_n → f$ uniformly on $S$ and each $f_n$ is cont on $S$. Let $(x_n)$ be a sequence of points in $S$ converging to $x \in S$. Then $f_n(x_n) → f(x)$.

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