TITLE: An intuitive explanation of the Taylor expansion
QUESTION [27 upvotes]: Could you provide a geometric explanation of the Taylor series expansion?

REPLY [2 votes]: Performing an $n$-th finite Taylor expansion can be thought of as making the approximation that the function's $n$-th derivative is constant.
Try it yourself: let $f$ be a $n$ times differentiable function whose $n$-th derivative is constant, and suppose you know the values of $f^{(i)}(0)$, $0\leq i\leq n$. By integrating repeatedly, you'll find that this uniquely determines $f$ and produces the formula for a Taylor's expansion.
Intuitively, I find it plausible that neglecting higher order derivatives of a function shouldn't cause too large of an error. Taylor's theorem confirms this intuition.