Matematické Fórum

stuart clark · 14. 12. 2017 13:19

Which one is greater of two numbers $300!$ and $100^{300}$

kerajs · 14. 12. 2017 18:40 — Editoval kerajs (14. 12. 2017 18:47)

Maybe:

$\ln 100^{300}=300 \ln 100 =600 \ln 10\approx ....$
$\ln 300!=\sum_{i=1}^{300} \ln i \\ \int_{2}^{300} \ln (x- 1) dx < \sum_{i=1}^{300} \ln i <\int_{1}^{300} \ln x dx \\ ...< \ln 300!< ...$

edison · 14. 12. 2017 18:57

Or use Calculator integrated in the Windows (ver 7 or older):
100^300 = 1e+600
300! = 3.0605e+614
:-)

vanok · 15. 12. 2017 13:04 — Editoval vanok (15. 12. 2017 20:23)

Hi ↑ stuart clark:,

Personally I have shall demonstrate, for example, more generally that

$n!>(\frac n3)^n$ for $n>6$ , n natural number.

Autor skryl část textu. Zobrazit/skrýt!

Remark. There is several of the other elementary proofs without the formula of Stirling.

kerajs · 15. 12. 2017 14:27 — Editoval kerajs (15. 12. 2017 14:34)

Stirlingův vzorec (Stirling's approximation) :

$300!\approx \sqrt{2\pi \cdot 300}(\frac{300}{e})^{300}= 100^{300}\cdot \sqrt{2\pi \cdot 300}(\frac{3}{e})^{300}>100^{300}$

stuart clark · 19. 01. 2018 14:35

Thanks ↑ vanok:↑ kerajs:

Matematické Fórum

#1 14. 12. 2017 13:19

Greater of two numbers

#2 14. 12. 2017 18:40 — Editoval kerajs (14. 12. 2017 18:47)

Re: Greater of two numbers

#3 14. 12. 2017 18:57

Re: Greater of two numbers

#4 15. 12. 2017 13:04 — Editoval vanok (15. 12. 2017 20:23)

Re: Greater of two numbers

#5 15. 12. 2017 14:27 — Editoval kerajs (15. 12. 2017 14:34)

Re: Greater of two numbers

#6 19. 01. 2018 14:35

Re: Greater of two numbers

Zápatí