Question

How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?

score 12.6k · Answer 1 · 2010-10-12 01:40:22

12.6k

Views

How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?

Published on 12 Oct 2010 - 1:40

score 8.1k · Answer 2 · 2026-04-02 09:12:20

8.1k

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$n!+1$ being a perfect square

Published on 02 Apr 2026 - 9:12

score 8.2k · Answer 3 · 2026-04-08 13:28:27

8.2k

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Is there a limit of $\cos(n!)$?

Published on 08 Apr 2026 - 13:28

score 54.3k · Answer 4 · 2010-11-27 14:08:20

54.3k

Views

The product of $n$ consecutive integers is divisible by $n$ factorial

Published on 27 Nov 2010 - 14:08

score 20.6k · Answer 5 · 2010-11-27 14:33:38

20.6k

Views

The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)

Published on 27 Nov 2010 - 14:33

score 11.3k · Answer 6 · 2010-11-27 20:02:46

11.3k

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Factorial of a non-integer number

Published on 27 Nov 2010 - 20:02

score 17.6k · Answer 7 · 2026-04-02 09:24:10

17.6k

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Can $n!$ be a perfect square when $n$ is an integer greater than $1$?

Published on 02 Apr 2026 - 9:24

score 499 · Answer 8 · 2010-12-01 21:35:26

499

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Series inequality $\sum _{k=n}^{\infty } \frac{1}{k!}\leq \frac{2}{n!}$

Published on 01 Dec 2010 - 21:35

score 287 · Answer 9 · 2010-12-23 07:54:30

287

Views

Show that $\Pi_{i < j} (v_i - v_j) \le k^{n^2}$ for $1 \le v_1 < v_2 < ... < v_n = k$

Published on 23 Dec 2010 - 7:54

score 2.7k · Answer 10 · 2026-04-01 18:30:32

2.7k

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How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?

Published on 01 Apr 2026 - 18:30

score 3.2k · Answer 11 · 2011-01-20 07:46:30

3.2k

Views

Number of zeros not possible in $n!$

Published on 20 Jan 2011 - 7:46

score 11.8k · Answer 12 · 2011-01-22 22:42:02

11.8k

Views

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$

Published on 22 Jan 2011 - 22:42

score 1.7k · Answer 13 · 2011-01-23 22:36:21

1.7k

Views

How do I describe the growth of something that scales by a factorial?

Published on 23 Jan 2011 - 22:36

score 500 · Answer 14 · 2011-01-24 23:54:05

500

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Is this a new formula?

Published on 24 Jan 2011 - 23:54

score 3.4k · Answer 15 · 2011-02-01 22:45:00

3.4k

Views

Logarithm base 2 and factorials

Published on 01 Feb 2011 - 22:45

List Question

How to prove $a^n < n!$ for all $n$ sufficiently large, and $n! \leq n^n$ for all $n$, by induction?

$n!+1$ being a perfect square

Is there a limit of $\cos(n!)$?

The product of $n$ consecutive integers is divisible by $n$ factorial

The product of $n$ consecutive integers is divisible by $ n!$ (without using the properties of binomial coefficients)

Factorial of a non-integer number

Can $n!$ be a perfect square when $n$ is an integer greater than $1$?

Series inequality $\sum _{k=n}^{\infty } \frac{1}{k!}\leq \frac{2}{n!}$

Show that $\Pi_{i < j} (v_i - v_j) \le k^{n^2}$ for $1 \le v_1 < v_2 < ... < v_n = k$

How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes?

Number of zeros not possible in $n!$

Summation involving a factorial: $1 + \sum_{j=1}^{n} j!j$

How do I describe the growth of something that scales by a factorial?

Is this a new formula?

Logarithm base 2 and factorials

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