Q1029 (Q1029): Difference between revisions
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Created claim: relevant answers (P14): 3142760 |
Changed claim: relevant answers (P14): 3142760 |
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| Property / relevant answers: 3142760 / qualifier | |||
Defining formula: \lim_{n\rightarrow \infty}\sqrt[n]{\frac{(27)^n(n!)^3}{(3n)!}} = 1 | |||
![{\displaystyle \lim _{n\rightarrow \infty }{\sqrt[{n}]{\frac {(27)^{n}(n!)^{3}}{(3n)!}}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09142faa0240c5220e80e6f35b24b24b647eb310)
Revision as of 12:30, 29 April 2020
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Statements
B.8
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q_49
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![{\displaystyle {\sqrt[{n}]{\frac {(27)^{n}(n!)^{3}}{(3n)!}}}\sim \left({\frac {(27)^{n}\left({\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}\right)^{3}}{{\sqrt {6\pi n}}\left({\frac {3n}{e}}\right)^{3n}}}\right)^{\frac {1}{n}}={\frac {27{\sqrt {2\pi }}^{\frac {3}{n}}n^{\frac {3}{2n}}\left({\frac {n}{e}}\right)^{3}}{{\sqrt {6\pi }}^{\frac {1}{n}}n^{\frac {1}{2n}}\left({\frac {3}{e}}\right)^{3}n^{3}}}=\left({\frac {2\pi n}{\sqrt {3}}}\right)^{\frac {1}{n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/089ac5e563dfadb333cd1eee6030950ce63506c3)