Factorials Algebra

Question

I have the following inequality $$5*10^{-10} \geq \dfrac{2^{n+1}}{(n+1)!}$$

Is there any way this can be solved algebraically? If not, is there a method that is better than guessing, for finding the smallest $n$ which satisfies the inequality?

Answer 1

this is a method from which you can calculate the value of $n$ , but it need not be the most accurate , $\frac{2^{n+1}}{(n+1)!}$<2*($\frac{2}{3})^n$<$5*10^{-10}$ now using log you can calculate the value of $n$ , but note that is not the smallest $n$

Answer 2

It might help to think of this as a ratio to get a (very) rough estimate:

When does the factorial have 10 times as many digits as 2^n?

Factorial grows a bit faster than one digit per n (for small n), 2^n grows a bit faster than three digits per 10 n (2^10 = 1024). Then:

$ n = \frac{3}{10}n + 10 $

estimate is n=14 (not n+1, as in the original equation).

Or you could just count the number of digits, which shows n ~ 18

n            factorial     digits         2^n  digits  difference
-----------------------------------------------------------------
0                       1     1              1    1         0
1                       1     1              2    1         0
2                       2     1              4    1         0
3                       6     1              8    1         0
4                      24     2             16    2         0
5                     120     3             32    2         1
6                     720     3             64    2         1
7                    5040     4            128    3         1
8                   40320     5            256    3         2
9                  362880     6            512    3         3
10                3628800     7           1024    4         3
11               39916800     8           2048    4         4
12              479001600     9           4096    4         5
13             6227020800    10           8192    4         6
14            87178291200    11          16384    5         6
15          1307674368000    13          32768    5         8
16         20922789888000    14          65536    5         9
17        355687428096000    15         131072    6         9
18       6402373705728000    16         262144    6        10
19     121645100408832000    18         524288    6        12
20    2432902008176640000    19        1048576    7        12