Taylor series

A Taylor series approximates a function near a point using derivatives at that point. Around $x=a$,

$$f(x)=\sum _{n=0}^{\infty }\frac{f^{(n)}(a)}{n!}(x-a{)}^n$$

when the series converges to the function. The finite version is the Taylor polynomial; it is often the practical object used in calculations.

Around $a=0$ it is called a Maclaurin series:

$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots ,$$ $$\sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots .$$

Why it matters

Taylor series turn hard functions into polynomials, which are easier to differentiate, integrate, and approximate numerically. In physics, they justify small-angle approximations such as $\sin \theta \approx \theta$, and local linearisation around equilibria in mechanics and field theory.

Remainder and error

The point is not just the polynomial; it is also the error. A common form is

$$f(x)=T_n(x)+R_n(x),$$

where $R_n$ is the remainder. If $R_n\to 0$, the Taylor series represents the function in that region.

Common pitfalls

• Assuming every smooth function equals its Taylor series everywhere.
• Forgetting the expansion point $a$; a Taylor series is local.
• Using a small-angle approximation outside the small-angle regime.

Big-picture intuition

Taylor’s theorem says smooth functions often look polynomial under a microscope. The zeroth-order term is the value, the first-order term is the tangent-line approximation, and higher-order terms encode curvature and finer local shape. This is the basis of perturbation theory: solve an easy nearby problem, then add controlled correction terms.

See also Derivative, Differentiation rules, and Limits and continuity.