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Australopithecus

  • 3 years ago

find the Mclaurin series for f(x) = (1-x)^(-2) So I took the derivative of f(x) = (1-x)^(-2) f(x) = (1-x)^(-2) f'(x) = 2(1-x)^(-3) f''(x) = 6(1-x)^(-4) f'''(x) = 24(1-x)^(-5) f''''(x) = 120(1-x)^(-6) then I took f(0) for all the derivatives f(0) = 1 f''(0) = 2 f'''(0) = 6 f''''(0) = 24 f'''''(0) = 120 Can anyone show me how to get a series from this?

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  1. 91
    • 3 years ago
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    I think there is easier way but let me study it for a bit

  2. Australopithecus
    • 3 years ago
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    I assume I use the formula \[\frac{f^{n}(0)(x)^{n}}{n!}\]

  3. 91
    • 3 years ago
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    I will teach you a better way

  4. 91
    • 3 years ago
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    now , it is common knowlege(back of the book) \[\frac{1}{1-x}=\sum _{n=0}^{\infty } x^n\]

  5. Australopithecus
    • 3 years ago
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    I need to use Mclaurin series, I know that method at least I think I do I still need to seek help to ensure I'm doing it correclty

  6. Australopithecus
    • 3 years ago
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    though

  7. 91
    • 3 years ago
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    you want to find series for \[\frac{1}{(1-x)^2}\]

  8. Australopithecus
    • 3 years ago
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    yes but I have to use the Mclaurin series method

  9. 91
    • 3 years ago
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    notice \[\frac{D}{\text{Dx}}\left[\frac{1}{(1-x)}\right]=\frac{1}{(1-x)^2}\]

  10. eliassaab
    • 3 years ago
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    I think @91 is leading you to the easiest method.

  11. 91
    • 3 years ago
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    remember \[\frac{1}{(1-x)}=\sum _{n=0}^{\infty } x^n\] so take derivative of other side to get \[\frac{1}{(1-x)^2}=\sum _{n=0}^{\infty } n x^{n-1}\]

  12. Australopithecus
    • 3 years ago
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    this question is more about learning how to use the Mclaurin series method than anything else

  13. eliassaab
    • 3 years ago
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    We should add that the convergence is only valid for |x|<1

  14. Australopithecus
    • 3 years ago
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    I know the method he is using

  15. eliassaab
    • 3 years ago
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    Since the McLaurin Series is unique. It does not matter which method to use to find it.

  16. Australopithecus
    • 3 years ago
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    ugh, I should have asked a question with a trig function or e^(x) something

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