Taylor Polynomials and decreasing terms

[HOLALO]

Hi, I have a question about taylor polynomials.

https://wikimedia.org/api/rest_v1/media/math/render/svg/09523585d1633ee9c48750c11b60d82c82b315bfI 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.

 

[jambaugh]

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.