Question

Evaluating $\lim_{n\to\infty}\sum_{r=0}^n(-1)^r\binom{n}{2r}\left(\frac{x}{n}\right)^{2r}$

score 193 · Answer 1 · 2026-04-15 07:16:24

193

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Evaluating $\lim_{n\to\infty}\sum_{r=0}^n(-1)^r\binom{n}{2r}\left(\frac{x}{n}\right)^{2r}$

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score 115 · Answer 2 · 2026-04-12 17:59:39

115

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179

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score 51 · Answer 4 · 2026-04-14 18:00:51

51

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score 848 · Answer 5 · 2026-04-15 04:36:13

848

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Sum of First k binomial coefficients divided by 2^n

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score 1.2k · Answer 6 · 2026-04-14 19:06:16

1.2k

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score 124 · Answer 7 · 2026-04-10 06:13:56

124

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score 419 · Answer 8 · 2026-04-07 22:23:15

419

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score 1.2k · Answer 9 · 2026-04-13 18:17:23

1.2k

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Use mathematical induction to prove that the sum of the entries of the $k^{th}$ row of Pascal’s Triangle is $2^k$.

Published on 13 Apr 2026 - 18:17

score 545 · Answer 10 · 2026-04-14 19:03:29

545

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Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$

Published on 14 Apr 2026 - 19:03

score 169 · Answer 11 · 2026-04-14 21:28:19

169

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Asymptotics for $k^c +k - 1 \choose k$ when $c\geq 1$

Published on 14 Apr 2026 - 21:28

score 80 · Answer 12 · 2026-04-15 06:41:17

80

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Prove that $U_{a+b}=U_aU_b+U_{a-1}U_{b-1}$ for $a\ge1$ and $b\ge1$

Published on 15 Apr 2026 - 6:41

score 3.6k · Answer 13 · 2026-04-13 16:22:21

3.6k

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Show $\sum_{k=m}^n {k \choose r} = {n+1 \choose r+1} - {m \choose r+1}$

Published on 13 Apr 2026 - 16:22

score 700 · Answer 14 · 2026-03-27 11:32:19

700

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How to compute quintinomial coefficients?

Published on 27 Mar 2026 - 11:32

score 144 · Answer 15 · 2026-04-14 21:19:15

144

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When is $m! > \binom{n}{m}$?

Published on 14 Apr 2026 - 21:19

List Question

Evaluating $\lim_{n\to\infty}\sum_{r=0}^n(-1)^r\binom{n}{2r}\left(\frac{x}{n}\right)^{2r}$

Combinatorics and limit problem

Finding closed form for double summation binomial

Asymptotic behavior of the number of idempotent maps

Sum of First k binomial coefficients divided by 2^n

How to show that $(n-k)!n^{\underline{k}}=\Gamma(n+1)$

How do I expand a binomial coefficient into a closed form function?

How can I prove the sum of squares and the sum of cubes with binomial coefficients?

Use mathematical induction to prove that the sum of the entries of the $k^{th}$ row of Pascal’s Triangle is $2^k$.

Find the value of $\binom{2000}{2} + \binom{2000}{5} + \binom{2000}{8} + \cdots \binom{2000}{2000}$

Asymptotics for $k^c +k - 1 \choose k$ when $c\geq 1$

Prove that $U_{a+b}=U_aU_b+U_{a-1}U_{b-1}$ for $a\ge1$ and $b\ge1$

Show $\sum_{k=m}^n {k \choose r} = {n+1 \choose r+1} - {m \choose r+1}$

How to compute quintinomial coefficients?

When is $m! > \binom{n}{m}$?

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