If at a point , a function has a power series expansion
the coefficients are given by:
where is the nth derivative of evaluated at the point . Named after the English mathematician Brook Taylor, this infinite series is called the Taylor expansion of the function at .
The Taylor expansion of the exponential function is:
from which it follows that:
Taylor approximations require both an expression and a point around which to expand.
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Thus, around the point the polynomial behaves like .
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The derivative of the arctan function has singularities at and . Therefore, the radius of convergence of the Taylor approximation around the origin is .
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You can compute and visualize Taylor approximations using the TaylorApproximationTutor command.
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