Question

The limit $\lim_{n \to \infty} \int_0^1 f(\{n x\}) g(\{n/x\}) \, \mathrm{d} x$

score 208 · Answer 1 · 2021-12-13 18:26:49

208

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The limit $\lim_{n \to \infty} \int_0^1 f(\{n x\}) g(\{n/x\}) \, \mathrm{d} x$

Published on 13 Dec 2021 - 18:26

score 235 · Answer 2 · 2021-12-14 00:03:10

235

Views

Studying convergence of the series involving fractional part

Published on 14 Dec 2021 - 0:03

score 188 · Answer 3 · 2022-01-05 16:33:14

188

Views

Find the sum/asymptotic as $N \rightarrow \infty$ of $\sum_{a = a_1}^{a_2} \big\{{\frac{{a}^{2}-N}{2\, a}}\big\}$

Published on 05 Jan 2022 - 16:33

score 33 · Answer 4 · 2026-03-25 13:55:19

33

Views

Integral of factional part

Published on 25 Mar 2026 - 13:55

score 314 · Answer 5 · 2022-04-07 18:18:25

314

Views

Are the [x] and {x} functions defined for negative numbers?

Published on 07 Apr 2022 - 18:18

score 61 · Answer 6 · 2022-04-21 23:35:39

61

Views

Given irrational numbers $0<\gamma_1<\gamma_2<1,$ real numbers $0\leq a<b\leq 1,$ does $\exists n$ such that $\{n\gamma_1\}, \{n\gamma_2\}\in(a,b)?$

Published on 21 Apr 2022 - 23:35

score 161 · Answer 7 · 2022-05-09 01:50:11

161

Views

Fractional part and greatest integer function

Published on 09 May 2022 - 1:50

score 203 · Answer 8 · 2022-05-10 18:10:00

203

Views

Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.)

Published on 10 May 2022 - 18:10

score 154 · Answer 9 · 2026-03-27 18:28:01

154

Views

Questions about density of $\left\{\ \left( k+ \frac{1}{2} \right)^n \right\}$ in $[0,1]$

Published on 27 Mar 2026 - 18:28

score 98 · Answer 10 · 2022-05-14 14:07:47

98

Views

Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$

Published on 14 May 2022 - 14:07

score 394 · Answer 11 · 2022-05-25 16:49:39

394

Views

Find $\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$ where $\{x\}$ means $x - \lfloor x \rfloor$.

Published on 25 May 2022 - 16:49

score 97 · Answer 12 · 2022-06-09 14:20:04

97

Views

When is $\{x\}=\frac{1}{x}$?

Published on 09 Jun 2022 - 14:20

score 88 · Answer 13 · 2022-06-10 17:37:36

88

Views

Solving $x^2-3-\operatorname{frac}(x)=0$

Published on 10 Jun 2022 - 17:37

score 96 · Answer 14 · 2022-08-12 13:02:52

96

Views

Fractional part of $2\sqrt{b^n(b^{n+1}-1)}$ dense in $(0,1)$

Published on 12 Aug 2022 - 13:02

score 55 · Answer 15 · 2022-09-12 13:08:02

55

Views

How to show that the expression can be made arbitrarily close to 0?

Published on 12 Sep 2022 - 13:08

List Question

The limit $\lim_{n \to \infty} \int_0^1 f(\{n x\}) g(\{n/x\}) \, \mathrm{d} x$

Studying convergence of the series involving fractional part

Find the sum/asymptotic as $N \rightarrow \infty$ of $\sum_{a = a_1}^{a_2} \big\{{\frac{{a}^{2}-N}{2\, a}}\big\}$

Integral of factional part

Are the [x] and {x} functions defined for negative numbers?

Given irrational numbers $0<\gamma_1<\gamma_2<1,$ real numbers $0\leq a<b\leq 1,$ does $\exists n$ such that $\{n\gamma_1\}, \{n\gamma_2\}\in(a,b)?$

Fractional part and greatest integer function

Show that $x+a-[x-b],a,b\in\mathbb R^+$, is a periodic function. Find its period. $([.]$ represents greatest integer function.)

Questions about density of $\left\{\ \left( k+ \frac{1}{2} \right)^n \right\}$ in $[0,1]$

Solve for x: $\lfloor x\rfloor \{\sqrt{x}\}=1$ where $\{x\}=x-\lfloor x\rfloor$

Find $\int_1^\infty \frac{\{x\}}{x(x+1)}dx,$ where $\{x\}$ means $x - \lfloor x \rfloor$.

When is $\{x\}=\frac{1}{x}$?

Solving $x^2-3-\operatorname{frac}(x)=0$

Fractional part of $2\sqrt{b^n(b^{n+1}-1)}$ dense in $(0,1)$

How to show that the expression can be made arbitrarily close to 0?

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