Question

What is the expectation of $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...$, where each $u$ is a uniformly random real number in $(0,1)$?

score 122 · Answer 1 · 2023-02-22 04:12:44

122

Views

What is the expectation of $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...$, where each $u$ is a uniformly random real number in $(0,1)$?

Published on 22 Feb 2023 - 4:12

score 104 · Answer 2 · 2023-02-22 11:25:16

104

Views

What is the probability that $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...>e$, where each $u$ is a uniformly random real number in $(0,1)$?

Published on 22 Feb 2023 - 11:25

score 80 · Answer 3 · 2023-02-23 19:04:36

80

Views

Computing infinite product of differences

Published on 23 Feb 2023 - 19:04

score 61 · Answer 4 · 2023-02-28 07:25:23

61

Views

Convergence of infinite product and its limit

Published on 28 Feb 2023 - 7:25

score 57 · Answer 5 · 2023-03-07 22:31:23

57

Views

The limit of the product of $\frac{n}{n-k}$ where $k$ is constant and $n\rightarrow\infty$

Published on 07 Mar 2023 - 22:31

score 60 · Answer 6 · 2023-03-08 17:18:17

60

Views

The product of a sequence of certain product of primes

Published on 08 Mar 2023 - 17:18

score 56 · Answer 7 · 2026-02-23 06:40:32

56

Views

Showing that the limit of an infinite product equals a sine function times a t term

Published on 23 Feb 2026 - 6:40

score 172 · Answer 8 · 2023-03-16 11:22:17

172

Views

Definite integral over an infinite product

Published on 16 Mar 2023 - 11:22

score 121 · Answer 9 · 2023-03-22 18:27:27

121

Views

Question on convergence of infinite product

Published on 22 Mar 2023 - 18:27

score 160 · Answer 10 · 2023-03-24 12:24:02

160

Views

Asymptotics for $ f(n) = \prod_{k=2}^{n} \sqrt[k-1]{k}$?

Published on 24 Mar 2023 - 12:24

score 37 · Answer 11 · 2026-03-25 07:49:32

37

Views

An entire function that only has simple zeroes at the positive squares

Published on 25 Mar 2026 - 7:49

score 122 · Answer 12 · 2023-03-27 23:14:24

122

Views

Is it possible to show that $\prod_{k=1}^{\infty} \left(1+\frac{1}{k^2}\right) = \frac{1}{2\pi}(e^\pi-e^{-\pi})$?

Published on 27 Mar 2023 - 23:14

score 51 · Answer 13 · 2026-03-25 07:44:51

51

Views

Ash - Complex Variables - Sufficient conditions for the absolute and uniform convergence of the infinite product of functions

Published on 25 Mar 2026 - 7:44

score 145 · Answer 14 · 2023-04-18 17:29:48

145

Views

Computing a limit involving the product

Published on 18 Apr 2023 - 17:29

score 69 · Answer 15 · 2023-04-29 21:00:21

How to prove $\displaystyle\prod_{n=1}^{\infty}n^{\frac{(-1)^{n+1}}{n^2}}=\left(\frac{A^{12}}{4\pi e^{\gamma}}\right)^{\eta(2)}$

List Question

What is the expectation of $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...$, where each $u$ is a uniformly random real number in $(0,1)$?

What is the probability that $(1+u_1)(1+u_1 u_2)(1+u_1 u_2 u_3)...>e$, where each $u$ is a uniformly random real number in $(0,1)$?

Computing infinite product of differences

Convergence of infinite product and its limit

The limit of the product of $\frac{n}{n-k}$ where $k$ is constant and $n\rightarrow\infty$

The product of a sequence of certain product of primes

Showing that the limit of an infinite product equals a sine function times a t term

Definite integral over an infinite product

Question on convergence of infinite product

Asymptotics for $ f(n) = \prod_{k=2}^{n} \sqrt[k-1]{k}$?

An entire function that only has simple zeroes at the positive squares

Is it possible to show that $\prod_{k=1}^{\infty} \left(1+\frac{1}{k^2}\right) = \frac{1}{2\pi}(e^\pi-e^{-\pi})$?

Ash - Complex Variables - Sufficient conditions for the absolute and uniform convergence of the infinite product of functions

Computing a limit involving the product

How to prove $\displaystyle\prod_{n=1}^{\infty}n^{\frac{(-1)^{n+1}}{n^2}}=\left(\frac{A^{12}}{4\pi e^{\gamma}}\right)^{\eta(2)}$

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