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Taylor's Theorem Approximation

  1. Jul 29, 2007 #1
    1. The problem statement, all variables and given/known data

    I need to use Taylor's thm to get an approximation to sqrt(5) with an error of no more than 2^(-9) and am totally lost.

    2. Relevant equations

    Taylor's theorem: Rn(x) = f(n)(y)/n! *x^n -- where f(n) is the nth derivative of f and Rn is R sub n.

    3. The attempt at a solution
  2. jcsd
  3. Jul 30, 2007 #2
    Are you sure you need Taylor's theorem?

    The most common way to obtain the square root of y is through the iteration

    Last edited: Jul 30, 2007
  4. Jul 30, 2007 #3
    Oh- ok, I can guess.

    You're probably after expanding


    The first few terms are 2+1/4-1/64+1/512+... (check!)

    There's probably a general pattern you can work out.
  5. Jul 30, 2007 #4

    Gib Z

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    Homework Helper

    It may be easier in this form:

    The Taylor Series, If it exists, of a function centered about the x coordinate a is given by: [tex]\sum_{n=0}^{\infty} \frac{ f^n (x-a)^n}{n!}[/tex].

    So in this let a=4, and also try and find a general pattern for the derivatives.

    We centered around 4 because the square root of that is just 2, and also because if we were to get more accurate by centering around 5, the derivatives would contain sqrt 5, which we are trying to find.
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