Question

Prove the identity $ \sum_{i=0}^n {n \choose i} i = n2^{n-1} $ with the identity $\sum_{k=0}^n \binom{n}{k}=2^n.$

score 103 · Answer 1 · 2016-09-14 03:22:09

103

Views

Prove the identity $ \sum_{i=0}^n {n \choose i} i = n2^{n-1} $ with the identity $\sum_{k=0}^n \binom{n}{k}=2^n.$

Published on 14 Sep 2016 - 3:22

score 76 · Answer 2 · 2016-09-16 20:06:36

76

Views

If $\sum_{n=1}^\infty a_n$and $\sum_{n=1}^\infty b_n$ are convergent and $a_n, b_n > 0$ then so is $\sum_{n=1}^\infty \max\lbrace a_n, b_n\rbrace$

Published on 16 Sep 2016 - 20:06

score 1.5k · Answer 3 · 2026-03-30 02:11:27

1.5k

Views

Prove that if $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$ then $\gcd(bc,ac,ab)=1$?

Published on 30 Mar 2026 - 2:11

score 318 · Answer 4 · 2026-03-31 08:40:56

318

Views

Euler's equation of variational calculus: is the proof of Gelfand and Fomin valid for non-analytic integrands?

Published on 31 Mar 2026 - 8:40

score 1k · Answer 5 · 2016-09-18 22:50:06

1k

Views

Nested Interval Property for open intervals and a alternative proof for NIP.

Published on 18 Sep 2016 - 22:50

score 339 · Answer 6 · 2016-09-20 15:29:31

339

Views

Proof without using Stone-Weierstrass theorem.

Published on 20 Sep 2016 - 15:29

score 8k · Answer 7 · 2016-09-21 13:38:20

8k

Views

Showing the standard topology on $\mathbb{R}$ has a countable basis

Published on 21 Sep 2016 - 13:38

score 124 · Answer 8 · 2016-09-22 07:27:36

124

Views

What natural numbers $k>2$ satisfy $k-2|2k$?

Published on 22 Sep 2016 - 7:27

score 227 · Answer 9 · 2016-09-22 09:35:58

227

Views

Could someone provide me with an elegant visual proof of the Taylor Series expansion?

Published on 22 Sep 2016 - 9:35

score 2.2k · Answer 10 · 2016-09-22 13:18:05

2.2k

Views

How to show the co-countable topology is a topology?

Published on 22 Sep 2016 - 13:18

score 365 · Answer 11 · 2026-03-27 16:21:12

365

Views

Approximation of Lipschitz function in uniform norm

Published on 27 Mar 2026 - 16:21

score 57 · Answer 12 · 2016-09-24 04:12:44

57

Views

Alternative proof for $\left|\frac{a}b+\frac{b}a\right|\ge 2$?

Published on 24 Sep 2016 - 4:12

score 145 · Answer 13 · 2016-09-24 16:16:43

145

Views

Let $R$ an ordered ring, with $a,b\ge 0$ and $a^n=b^n$ with $n\in\Bbb N_{>0}$. Prove that $b=a$

Published on 24 Sep 2016 - 16:16

score 274 · Answer 14 · 2026-03-27 01:47:45

274

Views

Prove that $a^{p^{k-1}(p-1)q^{l-1}(q-1)}\equiv 1 \pmod {p^kq^l}$

Published on 27 Mar 2026 - 1:47

score 337 · Answer 15 · 2026-03-25 12:53:49

337

Views

show that $ \limsup n\; | \;\{ (n+1)^2 \sqrt{2}\} - \{ n^2 \sqrt{2}\}\; | = \infty $

Published on 25 Mar 2026 - 12:53

List Question

Prove the identity $ \sum_{i=0}^n {n \choose i} i = n2^{n-1} $ with the identity $\sum_{k=0}^n \binom{n}{k}=2^n.$

If $\sum_{n=1}^\infty a_n$and $\sum_{n=1}^\infty b_n$ are convergent and $a_n, b_n > 0$ then so is $\sum_{n=1}^\infty \max\lbrace a_n, b_n\rbrace$

Prove that if $\gcd(a,b)=\gcd(b,c)=\gcd(a,c)=1$ then $\gcd(bc,ac,ab)=1$?

Euler's equation of variational calculus: is the proof of Gelfand and Fomin valid for non-analytic integrands?

Nested Interval Property for open intervals and a alternative proof for NIP.

Proof without using Stone-Weierstrass theorem.

Showing the standard topology on $\mathbb{R}$ has a countable basis

What natural numbers $k>2$ satisfy $k-2|2k$?

Could someone provide me with an elegant visual proof of the Taylor Series expansion?

How to show the co-countable topology is a topology?

Approximation of Lipschitz function in uniform norm

Alternative proof for $\left|\frac{a}b+\frac{b}a\right|\ge 2$?

Let $R$ an ordered ring, with $a,b\ge 0$ and $a^n=b^n$ with $n\in\Bbb N_{>0}$. Prove that $b=a$

Prove that $a^{p^{k-1}(p-1)q^{l-1}(q-1)}\equiv 1 \pmod {p^kq^l}$

show that $ \limsup n\; | \;\{ (n+1)^2 \sqrt{2}\} - \{ n^2 \sqrt{2}\}\; | = \infty $

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