What is the intuition behind the Taylor series?

The tangent line is the line that best approximates a given function at a given point, because its value and first derivative coincide with the function's.

The first two terms of the Taylor series is the best parabola approximating a given function at a point, because its value and first two derivatives coincide.

… and so on.

In the limit, the Taylor series is the best analytic function approximating a given function at a given point, because all the derivatives coincide.

Of course if the original function is analytic to begin with, like most of the functions you're likely to be familiar with, then the Taylor series is simply the original function.

Answer by Daniel McLaury:

The tangent line is the line that best approximates a given function at a given point, because its value and first derivative coincide with the function's.

The first two terms of the Taylor series is the best parabola approximating a given function at a point, because its value and first two derivatives coincide.

… and so on.

In the limit, the Taylor series is the best analytic function approximating a given function at a given point, because all the derivatives coincide.

Of course if the original function is analytic to begin with, like most of the functions you're likely to be familiar with, then the Taylor series is simply the original function.

What is the intuition behind the Taylor series?

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