Question

Is it possible to simplify the sum $\sum\limits_{n = 0}^{\lfloor {\frac{M}{2}}\rfloor } {\frac{{M!}}{{{2^n} \times ( {n!} )\times( {M - 2n})!}}}$?

score 67 · Answer 1 · 2020-09-29 14:17:18

Is it possible to simplify the sum $\sum\limits_{n = 0}^{\lfloor {\frac{M}{2}}\rfloor } {\frac{{M!}}{{{2^n} \times ( {n!} )\times( {M - 2n})!}}}$?

score 17 · Answer 2 · 2026-03-12 16:18:00

17

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score 54 · Answer 3 · 2020-10-03 08:34:27

54

Views

Simplifying $\frac{3 - 8}{\sqrt[3]{9} + 2\sqrt[3]{3} + 4} - \sqrt[3]{3}$. How to find the value?

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score 47 · Answer 4 · 2020-10-03 14:58:40

47

Views

Are there multiple ways to write some summations? (Example below)

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score 398 · Answer 5 · 2026-04-04 02:35:26

398

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score 115 · Answer 6 · 2020-10-05 14:07:12

Comparing $(2+\frac{1}{2})(3+\frac{1}{3})(4+\frac{1}{4})(5+\frac{1}{5})$ with $(2+\frac{1}{5})(3+\frac{1}{4})(4+\frac{1}{3})(5+\frac{1}{2})$

score 81 · Answer 7 · 2026-03-30 17:37:30

81

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score 305 · Answer 8 · 2020-10-08 05:51:22

305

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score 70 · Answer 9 · 2026-03-31 05:04:43

70

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How can $\frac{(-1)^{n+1}}{n^s} = \frac{1}{(2n-1)^s}-\frac{1}{(2n)^s}$?

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score 73 · Answer 10 · 2020-10-11 04:55:40

73

Views

How does $2\le1+2\cos x\le3$ become $\frac{2}{3}\le\frac{2}{1+2\cos x}\le1$?

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score 99 · Answer 11 · 2026-03-30 17:38:16

99

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score 27 · Answer 12 · 2020-10-14 17:30:41

27

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score 51 · Answer 13 · 2020-10-16 11:06:37

51

Views

Why does this hold?

Published on 16 Oct 2020 - 11:06

score 100 · Answer 14 · 2026-02-23 11:11:50

100

Views

Find minimum $n$ that satisfies $\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=\frac{12}{13}$

Published on 23 Feb 2026 - 11:11

score 121 · Answer 15 · 2026-04-03 20:14:52

121

Views

A model that can be followed when solving Diophantine equations - ideas?

Published on 03 Apr 2026 - 20:14

List Question

Is it possible to simplify the sum $\sum\limits_{n = 0}^{\lfloor {\frac{M}{2}}\rfloor } {\frac{{M!}}{{{2^n} \times ( {n!} )\times( {M - 2n})!}}}$?

Given a fraction with $ x $ is a real number, judge the scope of $ x $.

Simplifying $\frac{3 - 8}{\sqrt[3]{9} + 2\sqrt[3]{3} + 4} - \sqrt[3]{3}$. How to find the value?

Are there multiple ways to write some summations? (Example below)

How to replace floor to ceil

Comparing $(2+\frac{1}{2})(3+\frac{1}{3})(4+\frac{1}{4})(5+\frac{1}{5})$ with $(2+\frac{1}{5})(3+\frac{1}{4})(4+\frac{1}{3})(5+\frac{1}{2})$

Can generalised continued fractions always be written so that they are simple?

Tricks for questions involving finding integer solutions for a fraction which contains variables

How can $\frac{(-1)^{n+1}}{n^s} = \frac{1}{(2n-1)^s}-\frac{1}{(2n)^s}$?

How does $2\le1+2\cos x\le3$ become $\frac{2}{3}\le\frac{2}{1+2\cos x}\le1$?

Continued fraction: Closed form solution

Calculate the value of $x$

Why does this hold?

Find minimum $n$ that satisfies $\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=\frac{12}{13}$

A model that can be followed when solving Diophantine equations - ideas?

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