Taylor Series

The central idea

The Taylor series of $f$ centered at $a$ is $\sum_{n=0}^{\infty}f^{(n)}(a)(x-a)^n/n!$. It represents $f$ at a point only when the Taylor remainder tends to zero there. A series can converge while converging to something other than the originating smooth function.

Definitions, hypotheses, and notation

Taylor series may be manipulated within their common convergence interiors. Substitution changes both powers and domain: replacing $x$ by $g(x)$ requires $g(x)$ to lie in the base series interval. Multiplication uses coefficient convolution, while differentiation and integration preserve radius but can alter endpoints.

For alternating series with decreasing magnitudes, the first omitted term bounds truncation error. The Lagrange remainder is more general but requires a bound on a higher derivative. Choosing the simpler valid bound is part of method selection, not an afterthought.

Conceptual meaning

The infinite coefficient list records all derivative data at one center. When the remainder vanishes, these local data reconstruct the function throughout a convergence region and make algebraic approximation possible.

A dependable method and decision rule

1. Find derivative patterns or transform a known series.
2. Write the general coefficient with center and factorial.
3. Determine the radius and test endpoints.
4. Justify equality to the function using a remainder theorem or known identity.
5. For approximation, truncate and bound the tail.

Fully worked example

Common mistakes and why they fail