Question

Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.

score 913 · Answer 1 · 2026-04-03 14:30:59

913

Views

Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.

Published on 03 Apr 2026 - 14:30

score 287 · Answer 2 · 2022-10-28 16:57:37

287

Views

Prove $\prod\limits_{n=1}^\infty (1 + a_n)$ converges if and only if $\sum\limits_{n=1}^\infty a_n$ converges

Published on 28 Oct 2022 - 16:57

score 224 · Answer 3 · 2026-03-26 11:18:10

224

Views

What is $x^\bot$? Is $\zeta(\bot)=\bot$ for Riemann's zeta function $\zeta$ and wheel theory's $\bot$?

Published on 26 Mar 2026 - 11:18

score 136 · Answer 4 · 2026-03-29 16:02:10

136

Views

An infinite sum of products

Published on 29 Mar 2026 - 16:02

score 104 · Answer 5 · 2022-10-31 11:09:28

104

Views

Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?

Published on 31 Oct 2022 - 11:09

score 116 · Answer 6 · 2022-11-02 07:05:13

116

Views

Intuitive explanation for elegant property of the unit circle: product of lengths converges to $2^{-n}$.

Published on 02 Nov 2022 - 7:05

score 789 · Answer 7 · 2026-03-30 00:19:49

789

Views

Is my claim that this number is irrational correct?

Published on 30 Mar 2026 - 0:19

score 193 · Answer 8 · 2022-11-05 03:17:52

193

Views

Can $\prod\limits_{k=1}^\infty \left(1- \frac{1}{e^{ \sqrt{2} \pi k}}\right)$ be put into closed form?

Published on 05 Nov 2022 - 3:17

score 91 · Answer 9 · 2022-11-07 21:14:00

91

Views

Swapping the series and product in the zeta function euler product equality

Published on 07 Nov 2022 - 21:14

score 51 · Answer 10 · 2026-03-28 14:00:20

51

Views

Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that no sequence in $X$ converges to the constant function $1$.

Published on 28 Mar 2026 - 14:00

score 191 · Answer 11 · 2026-04-01 02:54:20

191

Views

What is the product of the areas of every regular polygon inscribed in a circle of area $1$?

Published on 01 Apr 2026 - 2:54

score 277 · Answer 12 · 2026-04-01 04:23:54

277

Views

Show that $\prod_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$

Published on 01 Apr 2026 - 4:23

score 93 · Answer 13 · 2022-11-13 10:42:56

93

Views

Stein complex analysis 6.2

Published on 13 Nov 2022 - 10:42

score 295 · Answer 14 · 2022-11-20 15:00:32

295

Views

Double infinite product $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$

Published on 20 Nov 2022 - 15:00

score 27 · Answer 15 · 2022-11-28 18:09:28

27

Views

Question on convergence to zero of an infinite product

Published on 28 Nov 2022 - 18:09

List Question

Prove or disprove: If $f(x)$ is continuous in $(0,1]$ and $f(x)\to\infty$ as $x\to 0^+$, then $\lim_{n\to\infty}\sum_{k=1}^n f(k/n)$ does not exist.

Prove $\prod\limits_{n=1}^\infty (1 + a_n)$ converges if and only if $\sum\limits_{n=1}^\infty a_n$ converges

What is $x^\bot$? Is $\zeta(\bot)=\bot$ for Riemann's zeta function $\zeta$ and wheel theory's $\bot$?

An infinite sum of products

Hidden property of the graph of $y=\tan{x}$: infinite product of lengths of zigzag line segments converges, but to what?

Intuitive explanation for elegant property of the unit circle: product of lengths converges to $2^{-n}$.

Is my claim that this number is irrational correct?

Can $\prod\limits_{k=1}^\infty \left(1- \frac{1}{e^{ \sqrt{2} \pi k}}\right)$ be put into closed form?

Swapping the series and product in the zeta function euler product equality

Let $X = \{ f \in \Bbb Q^\Bbb R : \sum_{x \in \Bbb R} |f(x)| < \infty \}$. Show that no sequence in $X$ converges to the constant function $1$.

What is the product of the areas of every regular polygon inscribed in a circle of area $1$?

Show that $\prod_{k=1}^\infty \frac{2k+1}{2\pi}\sin{\left(\frac{2\pi}{2k+1}\right)}\sec{\left(\frac{\pi}{k+2}\right)}=\frac{\pi}{2}$

Stein complex analysis 6.2

Double infinite product $\prod_\limits{0<i<j<\infty} \left\{i^{\frac{1}{i}} - j^{\frac{1}{j}}\right\}$

Question on convergence to zero of an infinite product

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