Question

Proof Verification: $\binom{n}{k}$ is always a natural number if $0\leq k\leq n$ - Spivak's Calculus, Problem 2-3(b)

score 86 · Answer 1 · 2019-07-28 12:49:35

86

Views

Proof Verification: $\binom{n}{k}$ is always a natural number if $0\leq k\leq n$ - Spivak's Calculus, Problem 2-3(b)

Published on 28 Jul 2019 - 12:49

score 256 · Answer 2 · 2026-03-26 06:21:23

256

Views

Compute $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^42^k{2k \choose k}}$

Published on 26 Mar 2026 - 6:21

score 74 · Answer 3 · 2026-03-26 16:10:06

74

Views

How to calculate the sum of a finite series: $P(k) = \sum_{i=1}^{k} S_2(m,i) \frac{n!}{n^m (n-i)!}$?

Published on 26 Mar 2026 - 16:10

score 371 · Answer 4 · 2026-03-27 13:24:24

371

Views

Finding $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^n 2^k {2k \choose k}}$

Published on 27 Mar 2026 - 13:24

score 320 · Answer 5 · 2019-07-29 17:31:40

320

Views

Sums of binomial coefficients in which lower index remains fixed

Published on 29 Jul 2019 - 17:31

score 382 · Answer 6 · 2026-03-25 16:02:51

382

Views

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

Published on 25 Mar 2026 - 16:02

score 91 · Answer 7 · 2019-08-01 16:30:42

91

Views

Is there a generalization of Faulhaber's formula is sense of summation bounds?

Published on 01 Aug 2019 - 16:30

score 3.1k · Answer 8 · 2019-08-02 11:23:30

3.1k

Views

Is there a way to find the specific variable coefficient in a binomial expansion?

Published on 02 Aug 2019 - 11:23

score 395 · Answer 9 · 2026-03-26 17:53:30

395

Views

How many pairs $(A_1, A_2)$ of subsets of $\{1,2,\ldots,n\}$ are there such that $A_1 \cap A_2 = \emptyset$?

Published on 26 Mar 2026 - 17:53

score 532 · Answer 10 · 2026-03-27 14:56:47

532

Views

A closed form for the sum $\frac{a}{b}+\frac{a\cdot(a+1)}{b\cdot(b+1)}+\frac{a\cdot(a+1)\cdot(a+2)}{b\cdot(b+1)\cdot(b+2)}+\cdots$

Published on 27 Mar 2026 - 14:56

score 224 · Answer 11 · 2026-03-26 06:21:42

224

Views

The limit $\lim_{r\to0}\frac1r\left(1-\binom{n}{r}^{-1}\right)$

Published on 26 Mar 2026 - 6:21

score 64 · Answer 12 · 2019-08-04 18:33:14

64

Views

When do we have $(a-b)^n=(a+b)^{-n}$?

Published on 04 Aug 2019 - 18:33

score 248 · Answer 13 · 2019-08-05 20:13:08

248

Views

Gregory coefficients integral expressions

Published on 05 Aug 2019 - 20:13

score 121 · Answer 14 · 2026-03-27 23:35:26

121

Views

In the exapnsion of $(1+x+x^3+x^4)^{10}$, find the coefficient of $x^4$

Published on 27 Mar 2026 - 23:35

score 92 · Answer 15 · 2019-08-07 02:40:23

92

Views

zero's for odd and $\pm 1$ for even. Or $\pm 1$ for odd and zero's for even

Published on 07 Aug 2019 - 2:40

List Question

Proof Verification: $\binom{n}{k}$ is always a natural number if $0\leq k\leq n$ - Spivak's Calculus, Problem 2-3(b)

Compute $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^42^k{2k \choose k}}$

How to calculate the sum of a finite series: $P(k) = \sum_{i=1}^{k} S_2(m,i) \frac{n!}{n^m (n-i)!}$?

Finding $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^n 2^k {2k \choose k}}$

Sums of binomial coefficients in which lower index remains fixed

Challenging sum: Calculate $\sum_{k=1}^\infty\frac{(-1)^{k-1}}{k^52^k{2k \choose k}}$

Is there a generalization of Faulhaber's formula is sense of summation bounds?

Is there a way to find the specific variable coefficient in a binomial expansion?

How many pairs $(A_1, A_2)$ of subsets of $\{1,2,\ldots,n\}$ are there such that $A_1 \cap A_2 = \emptyset$?

A closed form for the sum $\frac{a}{b}+\frac{a\cdot(a+1)}{b\cdot(b+1)}+\frac{a\cdot(a+1)\cdot(a+2)}{b\cdot(b+1)\cdot(b+2)}+\cdots$

The limit $\lim_{r\to0}\frac1r\left(1-\binom{n}{r}^{-1}\right)$

When do we have $(a-b)^n=(a+b)^{-n}$?

Gregory coefficients integral expressions

In the exapnsion of $(1+x+x^3+x^4)^{10}$, find the coefficient of $x^4$

zero's for odd and $\pm 1$ for even. Or $\pm 1$ for odd and zero's for even

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