Maclaurin Series

The central idea

A Maclaurin series is a Taylor series centered at zero: $f(x)=\sum_{n=0}^{\infty}f^{(n)}(0)x^n/n!$ where the series converges to $f$. Standard expansions include $e^x=\sum x^n/n!$, $\sin x=\sum(-1)^nx^{2n+1}/(2n+1)!$, and $\cos x=\sum(-1)^nx^{2n}/(2n)!$.

Definitions, hypotheses, and notation

Known series can be substituted into, multiplied, differentiated, or integrated inside their convergence domains. For example, replacing $x$ by $x^2$ in the exponential series gives $e^{x^2}=\sum x^{2n}/n!$. Such transformations are usually faster and less error-prone than recomputing many derivatives.

Analyticity is stronger than having derivatives of every order. A smooth function can have a Maclaurin series that fails to reproduce it away from zero. Equality requires a remainder tending to zero, so convergence of the coefficient series alone is not the entire justification.

Conceptual meaning

The coefficients encode all derivatives at the origin. A partial sum matches the function's value, slope, curvature, and successively higher derivative data there, turning a transcendental function into a locally accurate polynomial.

A dependable method and decision rule

1. Compute derivatives and evaluate them at zero, or begin from a known base series.
2. Divide the $n$th derivative value by $n!$.
3. Track which powers vanish because of symmetry or derivative cycles.
4. State a radius or interval on which equality is justified.
5. Use a remainder estimate when a numerical accuracy claim is required.

Fully worked example

Common mistakes and why they fail