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amyna

  • one year ago

Find limit of a sequence. Please help i don't know where to start! (1-(1/n))^(n)

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  1. amyna
    • one year ago
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    |dw:1444271970378:dw|

  2. amyna
    • one year ago
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    and i haven't learned l'hopitals rule, so i can't use it in this problem

  3. freckles
    • one year ago
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    \[e^k=\lim_{n \rightarrow \infty}(1+\frac{k}{n})^n \text{ is a well known limit }\]

  4. amyna
    • one year ago
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    yes but how to i use it?

  5. freckles
    • one year ago
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    in place of k you have -1...

  6. amyna
    • one year ago
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    since its a minus instead of a plus

  7. freckles
    • one year ago
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    1-something is the same as 1+(-something)

  8. SolomonZelman
    • one year ago
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    |dw:1444272525064:dw|

  9. amyna
    • one year ago
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    so the answer would just be e^1

  10. SolomonZelman
    • one year ago
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    This is true \(\forall k\)

  11. freckles
    • one year ago
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    you mean e^(-1)?

  12. amyna
    • one year ago
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    ahhh okay i think i got it ! thanks guys! :)

  13. SolomonZelman
    • one year ago
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    You can demostrate this using Stirling's approximation for example.

  14. SolomonZelman
    • one year ago
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    well, stirling's approximation of n! I am talking about. (And consequentially it would work for only integers, but this is in fact true for all k)

  15. SolomonZelman
    • one year ago
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    just that for integers we got a demonstration. I won't post that, but x! ~x^x(√[2πx]) / e^x is the approximation of Stirling.

  16. SolomonZelman
    • one year ago
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    that you can use to re-write the limit and reduce it (to e^k, if you do it correctly)

  17. SolomonZelman
    • one year ago
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    bye

  18. amyna
    • one year ago
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    the formula is much easier. Thanks everyone!

  19. amyna
    • one year ago
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    how about this question: (1 + x/n)^3n what would i do to the exponent 3n?

  20. amyna
    • one year ago
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    |dw:1444273022800:dw|

  21. SolomonZelman
    • one year ago
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    \(\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}\left(1+\frac{x}{n}\right)^{3n}}\) \(\large\color{slate}{\displaystyle \left(\lim_{n \rightarrow ~\infty}\left(1+\frac{x}{n}\right)^{n}\right)^3}\)

  22. SolomonZelman
    • one year ago
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    you know the limit inside the ( ... )\(^3\)

  23. SolomonZelman
    • one year ago
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    amyna, you aren't replying.... what's the problem?

  24. amyna
    • one year ago
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    i don't get why the whole limit is cubed

  25. SolomonZelman
    • one year ago
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    Well, \(\Large \left(A\right)^{bc}=\left(A^b\right)^c\)

  26. amyna
    • one year ago
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    right, okay. then whats the next step?

  27. SolomonZelman
    • one year ago
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    well, what is: \(\large\color{slate}{\displaystyle\lim_{n \rightarrow ~\infty}\left(1+\frac{x}{n}\right)^{n}}\) can you tell me that?

  28. amyna
    • one year ago
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    e^x

  29. SolomonZelman
    • one year ago
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    yes, and when the whole thing is cubed, \(\large\color{red}{\displaystyle \left(\color{black}{\lim_{n \rightarrow ~\infty}\left(1+\frac{x}{n}\right)^{n}}\right)^3}\) then, what do you get?

  30. amyna
    • one year ago
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    e^x^3 ?

  31. SolomonZelman
    • one year ago
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    Yes \((e^x)^3\), or \(e^{3x}\)

  32. amyna
    • one year ago
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    ok so thats the answer! Thank you for your help! :)

  33. SolomonZelman
    • one year ago
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    You are welcome:)

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