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Math Help - What is the difference between a Taylor series and power series representation?

  1. #1
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    Question What is the difference between a Taylor series and power series representation?

    Hi,

    A question asks to find a Maclaurin series for 1/(1+4x)

    I know 1/(1+x) = 1 - x + x^2 - x^3 + x^4 - ...

    Therefore 1/(1+4x) = 1 - 4x + (4x)^2 - (4x)^3 + (4x)^4 - ... = Sum(0, inf) (-1)^n * 4^n * x^n

    And the answer key says that is correct. But I am confused, because isn't that just a power series representation? I thought Maclaurin and Taylor series representations had to be of the form [f^(n)(C)/n!]*(x-C)^n. There is no n! in my answer, so how is it a Maclaurin series representation?

    tl;dr: What is the difference between a power series and maclaurin series representation?

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  2. #2
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    Re: What is the difference between a Taylor series and power series representation?

    A MacLaurin series is a power series, with "C" equal to 0. A "power series" is any infinite sum of functions where the functions are powers of x- C. A Taylor's series is a power series associated to a given function by a specific formula.
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  3. #3
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    Re: What is the difference between a Taylor series and power series representation?

    More specifically, a Taylor Series is a Power Series representation of a function \displaystyle \begin{align*} f(x) \end{align*} and is equal to \displaystyle \begin{align*} \sum_{ k = 0 }^{\infty} \frac{f^{(k)}(c)}{k!} \left( x - c \right) ^k \end{align*}. If you can find a series representation of a function by some other means (such as analysing a geometric series), then this will be equal to a Taylor series of the function (though you may need to write in any terms that have a 0 coefficient).
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