Question

Do I need to prove $\lim_{\|P\|\to 0} L(f,P)=\sup_P L(f,P)$, and $\lim_{\|P\|\rightarrow 0} U(f,P)=\inf_P U(f,P)$? Or is it trivial?

score 749 · Answer 1 · 2021-02-12 05:28:03

749

Views

Do I need to prove $\lim_{\|P\|\to 0} L(f,P)=\sup_P L(f,P)$, and $\lim_{\|P\|\rightarrow 0} U(f,P)=\inf_P U(f,P)$? Or is it trivial?

Published on 12 Feb 2021 - 5:28

score 214 · Answer 2 · 2021-02-16 03:57:33

214

Views

Completing a proof on integrability: $f$ is Riemann integrable under the mesh definition iff $\sup L_\Gamma = \inf U_\Gamma$

Published on 16 Feb 2021 - 3:57

score 92 · Answer 3 · 2021-02-19 07:16:33

92

Views

Proof that $\sum_{k=1}^n \frac{k}{2^k} < 2 $ for all $n \geq 1$

Published on 19 Feb 2021 - 7:16

score 132 · Answer 4 · 2021-02-22 19:42:08

132

Views

Periodic function's integral

Published on 22 Feb 2021 - 19:42

score 508 · Answer 5 · 2021-02-24 14:28:43

508

Views

Show that $f$ is Riemann-Integrable

Published on 24 Feb 2021 - 14:28

score 66 · Answer 6 · 2026-03-29 11:07:03

66

Views

how to show convergence and evaluate $n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2}$

Published on 29 Mar 2026 - 11:07

score 1k · Answer 7 · 2021-02-25 16:11:14

1k

Views

Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$

Published on 25 Feb 2021 - 16:11

score 850 · Answer 8 · 2021-02-25 17:53:45

850

Views

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is integrable.

Published on 25 Feb 2021 - 17:53

score 1.1k · Answer 9 · 2021-02-26 16:44:10

1.1k

Views

show that $f$ is Riemann integrable on $[0,1]$

Published on 26 Feb 2021 - 16:44

score 84 · Answer 10 · 2021-02-26 18:26:29

84

Views

Asymptotics of $\sum_{i=0}^n e^{-i^2/n + i^a}$ as $n \to \infty$

Published on 26 Feb 2021 - 18:26

score 188 · Answer 11 · 2021-02-26 20:43:35

188

Views

Show that $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$ is Riemann integrable on $[0,1]$

Published on 26 Feb 2021 - 20:43

score 197 · Answer 12 · 2021-02-27 14:28:57

197

Views

Let $f(x)=\frac{1}{n}$ if $x=\frac{1}{n}$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable

Published on 27 Feb 2021 - 14:28

score 158 · Answer 13 · 2021-02-27 16:29:42

158

Views

Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if not. Show that $f$ is not Riemann integrable

Published on 27 Feb 2021 - 16:29

score 141 · Answer 14 · 2021-02-27 19:51:18

141

Views

Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not Riemann integrable on $[0,1]$

Published on 27 Feb 2021 - 19:51

score 70 · Answer 15 · 2021-02-28 22:14:36

70

Views

Inequality $\sum_{k=1}^n\frac{k^2}{n^3+n^2}\le\sum_{k=1}^n\frac{k^2}{n^3+k^2}\le\sum_{k=1}^n\frac{k^2}{n^3+1}$

Published on 28 Feb 2021 - 22:14

List Question

Do I need to prove $\lim_{\|P\|\to 0} L(f,P)=\sup_P L(f,P)$, and $\lim_{\|P\|\rightarrow 0} U(f,P)=\inf_P U(f,P)$? Or is it trivial?

Completing a proof on integrability: $f$ is Riemann integrable under the mesh definition iff $\sup L_\Gamma = \inf U_\Gamma$

Proof that $\sum_{k=1}^n \frac{k}{2^k} < 2 $ for all $n \geq 1$

Periodic function's integral

Show that $f$ is Riemann-Integrable

how to show convergence and evaluate $n\sum_{k=1}^{2n} \frac{e^{-n/k}}{k^2}$

Show that $f$ is Riemann Integrable on $\left[0,\pi/2\right]$

Let $f(x)=1$ if $x=0$ and $f(x)=0$ if $x>0$. Show that $f$ is integrable.

show that $f$ is Riemann integrable on $[0,1]$

Asymptotics of $\sum_{i=0}^n e^{-i^2/n + i^a}$ as $n \to \infty$

Show that $f(x)=x$ if $x\in\mathbf{Q}$ and $f(x)=1-x$ if $x \in\mathbf{R}/\mathbf{Q}$ is Riemann integrable on $[0,1]$

Let $f(x)=\frac{1}{n}$ if $x=\frac{1}{n}$ and $f(x)=0$ if not. Show that $f$ is Riemann integrable

Let $f(x)=2|x|+1$ if $x\in\mathbf{Q}$ and $f(x)=0$ if not. Show that $f$ is not Riemann integrable

Show that $f=\sin(x)$ if $x\in\mathbb{Q}$ and $f(x)=\cos(x)$ if $x\in\mathbb{R}\setminus\mathbb{Q}$ is not Riemann integrable on $[0,1]$

Inequality $\sum_{k=1}^n\frac{k^2}{n^3+n^2}\le\sum_{k=1}^n\frac{k^2}{n^3+k^2}\le\sum_{k=1}^n\frac{k^2}{n^3+1}$

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