Question

What is $\int\limits_{0}^{1} \left[x(1-x)\right]^m \, dx$ ($m$ positive integer)?

score 196 · Answer 1 · 2016-03-03 03:16:48

196

Views

What is $\int\limits_{0}^{1} \left[x(1-x)\right]^m \, dx$ ($m$ positive integer)?

Published on 03 Mar 2016 - 3:16

score 35 · Answer 2 · 2026-03-27 16:53:12

35

Views

Binomial Coefficient Explanaition

Published on 27 Mar 2026 - 16:53

score 115 · Answer 3 · 2016-03-04 17:43:22

115

Views

Prove closed form for $\sum_k (2k)^2\binom{2n}{k+n} $ and $\sum_ k(2k+1)^2\binom{2n+1}{k+n+1}$

Published on 04 Mar 2016 - 17:43

score 882 · Answer 4 · 2016-03-05 09:47:05

882

Views

If $x$ is very small, then prove the following:

Published on 05 Mar 2016 - 9:47

score 1.3k · Answer 5 · 2016-03-05 11:43:54

1.3k

Views

The number of terms in the expansion of a binomial

Published on 05 Mar 2016 - 11:43

score 2k · Answer 6 · 2016-03-05 11:56:53

2k

Views

Total number of non similar triangles which can be formed such that all the angles of the triangles are integers

Published on 05 Mar 2016 - 11:56

score 4k · Answer 7 · 2016-03-09 04:13:46

4k

Views

Shortcut for finding number of rational terms in $(a^{\frac{1}{p}}+b^{\frac{1}{q}})^n$

Published on 09 Mar 2016 - 4:13

score 268 · Answer 8 · 2016-03-10 21:07:22

268

Views

Use the binomial theorem to prove if $m\mid b - a$, then $m \mid b^n - a^n$.

Published on 10 Mar 2016 - 21:07

score 114 · Answer 9 · 2016-03-11 16:37:11

114

Views

Find the sum of the series .

Published on 11 Mar 2016 - 16:37

score 18.3k · Answer 10 · 2016-03-13 05:56:58

18.3k

Views

Binomial Theorem Proof by Induction

Published on 13 Mar 2016 - 5:56

score 714 · Answer 11 · 2016-03-14 11:15:53

714

Views

Binomial expansion and convergence

Published on 14 Mar 2016 - 11:15

score 141 · Answer 12 · 2016-03-15 18:01:26

141

Views

Prove $\sum_{0}^{n}(-1)^i\binom{n}{i} = 0$

Published on 15 Mar 2016 - 18:01

score 96 · Answer 13 · 2016-03-16 10:28:57

96

Views

find $\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$

Published on 16 Mar 2016 - 10:28

score 93 · Answer 14 · 2016-03-16 17:53:10

93

Views

Prove that $n\cdot(x+1)^{n-1} = \sum_{k=0}^{n}k\cdot\binom{n}{k}\cdot x^{k-1}$

Published on 16 Mar 2016 - 17:53

score 83 · Answer 15 · 2016-03-17 08:37:15

83

Views

find the value of $\int_0^1(C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k})dy$

Published on 17 Mar 2016 - 8:37

List Question

What is $\int\limits_{0}^{1} \left[x(1-x)\right]^m \, dx$ ($m$ positive integer)?

Binomial Coefficient Explanaition

Prove closed form for $\sum_k (2k)^2\binom{2n}{k+n} $ and $\sum_ k(2k+1)^2\binom{2n+1}{k+n+1}$

If $x$ is very small, then prove the following:

The number of terms in the expansion of a binomial

Total number of non similar triangles which can be formed such that all the angles of the triangles are integers

Shortcut for finding number of rational terms in $(a^{\frac{1}{p}}+b^{\frac{1}{q}})^n$

Use the binomial theorem to prove if $m\mid b - a$, then $m \mid b^n - a^n$.

Find the sum of the series .

Binomial Theorem Proof by Induction

Binomial expansion and convergence

Prove $\sum_{0}^{n}(-1)^i\binom{n}{i} = 0$

find $\frac{3}{6}+\frac{3\cdot5}{6\cdot9}+\frac{3\cdot5\cdot7}{6\cdot9\cdot12}+\cdots$

Prove that $n\cdot(x+1)^{n-1} = \sum_{k=0}^{n}k\cdot\binom{n}{k}\cdot x^{k-1}$

find the value of $\int_0^1(C(-y-1)\cdot\sum_{k=1}^{2016}\frac{1}{y+k})dy$

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