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klimenkov

  • 3 years ago

\[\lim_{n\rightarrow\infty}\frac{\sqrt[n]{n!}}{n}\]

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  1. myko
    • 3 years ago
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    maybe like this: \[\sqrt[n]{n!}=\sqrt[n]{n(n-1)(n-2)\cdots1} =\sqrt[n]{n}\sqrt[n]{n-1}\cdots \sqrt[n]{1}=1\] so limit is equal to 0

  2. myko
    • 3 years ago
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    @klimenkov

  3. klimenkov
    • 3 years ago
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    Are you sure that \[\lim_{n\rightarrow\infty}\sqrt[n]{n!}=1\]

  4. myko
    • 3 years ago
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    look the steps from my comment befor. It looks ok

  5. myko
    • 3 years ago
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    all the roots at the right hand side: \[\sqrt[n]{n}=\sqrt[n]{n-1}=\cdots=\sqrt[n]{1}=1\]

  6. myko
    • 3 years ago
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    so their product too

  7. klimenkov
    • 3 years ago
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    No, it's not ok. Because you multiply an infinite quantity or 1. As we know \(1^{\infty}={}?\).

  8. myko
    • 3 years ago
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    \[1^{\infty} =1*1*\cdots*1=1\]

  9. klimenkov
    • 3 years ago
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    Very nice. What can you say about this pretty limit? \[\lim_{n\rightarrow\infty}\left(1+\frac1n\right)^n\]It is \(1^{\infty}\).

  10. myko
    • 3 years ago
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    this happens when you talk about functions. The reason of indetermination of 1^infinity is because of that. But in this case there are no functions involved. That's my point

  11. myko
    • 3 years ago
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    in this case there is just number one multiplyed infinitly many times. And it happens after the limit was taken

  12. klimenkov
    • 3 years ago
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    Ok. What about this? \[\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac1n\]Is it 0?

  13. myko
    • 3 years ago
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    this is a harmonic series. It is not convergent

  14. myko
    • 3 years ago
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    your 1º question was about sequences

  15. klimenkov
    • 3 years ago
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    Look at the denominator carefully please. I hopr you will try to get what I'm saying.

  16. myko
    • 3 years ago
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    sry, but i don't

  17. klimenkov
    • 3 years ago
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    Can you find this? \[\lim_{n\rightarrow\infty}\sum_{k=1}^n\frac1n\]

  18. myko
    • 3 years ago
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    another way to try this: \[\lim \sqrt[n]{\frac{n!}{n^n}} = 0\]

  19. myko
    • 3 years ago
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    infinity

  20. klimenkov
    • 3 years ago
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    Can you show the way you solve it?

  21. myko
    • 3 years ago
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    I don't remmeber the formal proof of n!/n^n =0, but it's evident, if you try a few first terms of this sequence. There are some posts about it if you google a bit

  22. klimenkov
    • 3 years ago
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    @myko, see this and tell me what is your mistake? http://www.wolframalpha.com/input/?i=Limit+%28n!%29^%281%2Fn%29%2Fn+n-%3Einfinity

  23. TuringTest
    • 3 years ago
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    @mahmit2012 a little help here?

  24. mahmit2012
    • 3 years ago
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    |dw:1351370532404:dw|

  25. TuringTest
    • 3 years ago
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    very nice, my turn...

  26. mahmit2012
    • 3 years ago
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    |dw:1351370740820:dw|

  27. mahmit2012
    • 3 years ago
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    |dw:1351370823511:dw|

  28. mahmit2012
    • 3 years ago
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    |dw:1351370856590:dw|

  29. TuringTest
    • 3 years ago
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    \[{\sqrt[n]{n!}\over n}=\exp\left(\frac1n\ln(n!)-\ln n\right)=\exp\left({n\ln n-n+O(n)-n\ln n\over n}\right)\]\[=\exp(-1)=\frac1e\]

  30. mukushla
    • 3 years ago
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    *

  31. myko
    • 3 years ago
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    ya I was wrong. Here is another way to solve it. As we know root test is stronger than cuotient test, so the folowing inequality holds: \[\lim \inf \frac{a_{n+1}}{a_{n}}\leq\lim \inf \sqrt[n]{a_{n}}\leq \lim \sup \sqrt[n]{a_{n}} \leq \lim \sup\frac{a_{n+1}}{a_{n}}\] let \[a_{n} = \frac{n!}{n^{n}}\] then \[\frac{a_{n+1}}{a_{n}}=\frac{1}{(1+\frac{1}{n})^{n}}=\frac{1}{e}\] this means that \[\lim \sqrt[n]{a_{n}} = \frac{1}{e}\]

  32. myko
    • 3 years ago
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    @mahmit2012 \[\lim \frac{a_{n+1}}{a_{n}}=\lim \sqrt[n]{a_{n}}\] only if a_n is convergent, what is not implied in this question

  33. klimenkov
    • 3 years ago
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    Nice. But I have one more interesting method to find it. \[\lim_{n\rightarrow\infty} \frac{\sqrt[n]{n!}}{n}=\lim_{n\rightarrow\infty}\sqrt[n]{\frac{n!}{n^n}}=\lim_{n\rightarrow\infty}\sqrt[n]{\frac1n\cdot\frac2n\cdots\frac n n}=A\]\[\ln A=\lim_{n\rightarrow\infty}\frac1n(\ln\frac1n+\ln\frac2n+\ldots+\ln\frac n n)=\int_0^1\ln x dx=-1\]\[\lim_{n\rightarrow\infty} \frac{\sqrt[n]{n!}}{n}=A=e^{-1}=\frac1e\]

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