Question

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

score 91 · Answer 1 · 2015-06-19 17:58:21

91

Views

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

Published on 19 Jun 2015 - 17:58

score 61 · Answer 2 · 2026-04-03 00:47:45

61

Views

Eliminating a summation

Published on 03 Apr 2026 - 0:47

score 224 · Answer 3 · 2026-03-26 15:17:00

224

Views

An uncountable and closed subset of the Liouville Numbers

Published on 26 Mar 2026 - 15:17

score 128 · Answer 4 · 2015-06-21 19:23:13

128

Views

Closed form for $\sqrt {-1\sqrt {-2 \sqrt {-3 \sqrt {-4 \ldots}}}}$

Published on 21 Jun 2015 - 19:23

score 274 · Answer 5 · 2015-06-24 06:39:17

274

Views

Finding closed form for a generating function with different powers of $x$ in parameter

Published on 24 Jun 2015 - 6:39

score 1.1k · Answer 6 · 2026-04-01 02:04:29

1.1k

Views

Simplifying real part of hypergeometric function with complex parameters

Published on 01 Apr 2026 - 2:04

score 564 · Answer 7 · 2015-06-28 23:51:23

564

Views

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion

Published on 28 Jun 2015 - 23:51

score 144 · Answer 8 · 2015-06-29 20:18:26

144

Views

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

Published on 29 Jun 2015 - 20:18

score 410 · Answer 9 · 2026-04-03 02:29:49

410

Views

Can the recurrence $C_n = 2 C_1 C_{n-1} - C_{n-2}$ be expressed in a closed expression?

Published on 03 Apr 2026 - 2:29

score 852 · Answer 10 · 2026-03-26 06:27:52

852

Views

A closed form for the coefficients of Chebyshev polynomials

Published on 26 Mar 2026 - 6:27

score 75 · Answer 11 · 2015-07-02 01:45:37

75

Views

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

Published on 02 Jul 2015 - 1:45

score 1.9k · Answer 12 · 2026-03-29 19:11:05

1.9k

Views

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

Published on 29 Mar 2026 - 19:11

score 130 · Answer 13 · 2015-07-03 22:09:50

130

Views

How to Evaluate this Summation to Find a Closed Form

Published on 03 Jul 2015 - 22:09

score 59 · Answer 14 · 2026-04-02 11:27:23

59

Views

Multiplication of 2 sums that equal another multiplication of 2 sums

Published on 02 Apr 2026 - 11:27

score 484 · Answer 15 · 2015-07-04 02:45:03

484

Views

Closed form for a binomial series

Published on 04 Jul 2015 - 2:45

List Question

Prove that for any positive integer $n$ and $d$, $\sum_{k=0}^d 2^k\log_2(\frac{n}{2^k})=2^{d+1}\log_2(\frac{n}{2^{d-1}})-2-\log_2{n}$

Eliminating a summation

An uncountable and closed subset of the Liouville Numbers

Closed form for $\sqrt {-1\sqrt {-2 \sqrt {-3 \sqrt {-4 \ldots}}}}$

Finding closed form for a generating function with different powers of $x$ in parameter

Simplifying real part of hypergeometric function with complex parameters

Evaluating $\int{ \frac{x^n}{1 + x + \frac{x^2}{2} + \cdots + \frac{x^n}{n!}}}dx$ using Pascal inversion

closed form for $\int_{0}^{\infty}\frac{ \beta(a+ix,a-ix)}{\beta(b+ix,b-ix)}\frac{dx}{(b^2+x^2)}$

Can the recurrence $C_n = 2 C_1 C_{n-1} - C_{n-2}$ be expressed in a closed expression?

A closed form for the coefficients of Chebyshev polynomials

Proving the closed form for an infinite sum (related to Chebyshev polynomials)

How can I prove $\pi=e^{3/2}\prod_{n=2}^{\infty}e\left(1-\frac{1}{n^2}\right)^{n^2}$?

How to Evaluate this Summation to Find a Closed Form

Multiplication of 2 sums that equal another multiplication of 2 sums

Closed form for a binomial series

Trending Questions

Popular # Hahtags

Popular Questions