Theorem. Suppose that

is a power series such that the radius of convergence is r > 0 then

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Definition. Suppose that f is a function such that the ith derivative of f exists at the real number c for i = 1, 2, ..., n. Then the polynomial

is called the nth degree Taylor polynomial for f at x = c.

Note that the first degree Taylor polynomial for f at x = c is the equation of the tangent line at x = c.

Examples.

  1. f(x) = ex at x = 0.
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  2. f(x) = sin(x) at x = 0.
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  3. f(x) = cos(3x) at x = 0.
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  4. f(x) = ln(x) at x = 1.
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