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

Graphical or geometric meaning

Successive Taylor polynomials agree with the function over a widening visible region inside convergence, yet can swing sharply near or outside a boundary. Endpoint behavior is not visible from local derivative matching alone.

Common mistakes and why they fail

Verification and reasonableness checks

• Differentiate or integrate back to a known series.
• Evaluate both sides at the center.
• Use remainder bounds or alternating error estimates.

Coefficients alone do not prove equality

The formal Taylor coefficients at $a$ are $f^{(n)}(a)/n!$, but representation of $f(x)$ requires the remainder to approach zero. A convergent numerical series need not otherwise converge to the original function. Build familiar series by substitution, differentiation, or integration only within an established radius, and then reassess endpoints. Integrating the geometric series gives $\ln(1+x)=\sum_{n=1}^\infty(-1)^{n+1}x^n/n$ for $-1<x\le1$: it converges conditionally at $x=1$ and diverges at $x=-1$. For approximation, state the center, degree, and a justified error bound or next-term estimate. Those details distinguish an identity from a formal pattern.

Practice

1. Write the series for $1/(1+x)$.
2. Write the series for $\ln(1+x)$.
3. What must $R_n(x)$ do for equality?

Connections and next steps

• Taylor Polynomial
• Maclaurin Series
• Interval of Convergence