An intuitive explanation of the Taylor expansion Could you provide a geometric explanation of the Taylor series expansion?

Think of a Taylor series not as one entity but as a sequence of approximations. 
The first term gives a constant approximation: $f(x + h)$ is approximately $f(x).$ 
The first two terms give a linear approximation: $f(x + h)$ is approximately $f(x)$ plus a trend term, $h f'(x).$
The first three terms include a constant approximation, a linear trend, and a curvature term to account for the change in the linear trend: f(x + h) is approximately $f(x) + h f'(x) + h^2 \dfrac{f''(x)}{2}.$
Next you add a term to account for the change in the curvature, etc.
