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B Taylor Polynomials and decreasing terms

  1. Mar 15, 2017 #1
    Hi, I have a question about taylor polynomials.

    https://wikimedia.org/api/rest_v1/media/math/render/svg/09523585d1633ee9c48750c11b60d82c82b315bf


    I was looking for proof that why every lagrange remainder is decreasing as the order of lagnrange remainder increases.

    so on wikipedia, it says, for a function to be an analytic function, x must be in the neighborhood of x0. What does this neighborhood mean by? should that be r=|x-x0|<1? then everything makes sense.
     
  2. jcsd
  3. Mar 16, 2017 #2

    jambaugh

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    A(n open) neighborhood of a point is an *open* set containing that point. If you need to pick some neighborhood then open disks or balls around the point work nicely. i.e. { x : |x - x_o=| < epsilon}. You can define "closed neighborhoods" as the closures of open neighborhoods. The defining property of neighborhoods is that sequences of points outside the neighborhood cannot get arbitrarily close to the point within it.
     
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