Taylor series

Let be times differentiable near . The Taylor polynomial of order is:

At , this is a Maclaurin expansion. If the series converges to :

this is the Taylor series of around .

Intuition

A Taylor series replaces a complicated function locally with a polynomial whose coefficients are fixed by derivatives at a point.

Order 0: constant value .
Order 1: linear approximation (tangent) — see Derivatives, gradient, and Hessian.
Order 2: quadratic approximation (curvature via the Hessian).
Higher order: better accuracy near , but often useless or unstable far from .

Central idea: any sufficiently smooth function behaves almost like a polynomial at small scale.

Univariate and multivariate cases

Expansion around up to order 2:

The linear term dominates for small ; the quadratic term captures curvature.

For near :

This is the order-2 Taylor expansion — foundation of Newton's method and delta-gamma approximations in finance.

Classic expansions

Function	Expansion (Maclaurin, small )	Usage
		Log-likelihood, stochastic processes
		Log returns, entropy
	Alternating series in even/odd powers	Signals, Fourier (out of scope here)
		Return approximations, binomial series
		Geometric series, discrete discounting
(normal)	Expansion around 0 or tail asymptotics	VaR, default probabilities (approx.)

Remainder and accuracy

Between and , there exists such that:

The remainder depends on the -th derivative on the interval — hence the importance of regularity.

We often write:

In practice: truncating at order 1 or 2 suffices for optimization and sensitivities; higher orders serve mainly in numerical analysis or statistical asymptotics.

Applications

Optimization — Newton's method: solve by linearizing the gradient (order-1 Taylor on ). Convergence analysis: quadratic rate of Newton near a simple root.

Finance — Delta-gamma: portfolio options P&L approximation: (order-2 Taylor of price). Duration-convexity (bonds): same logic for price vs yield.

Statistics — Taylor expansion around the MLE to approximate likelihood and derive tests (normal asymptotics). Approximation for small returns.

Numerical methods — Integration, root finding, finite differences — see Numerical methods (coming soon).

Limits and pitfalls

Situation	Issue	Consequence
Radius of convergence	Series diverges beyond some	Approximation valid only locally
Non- function	E.g. , ReLU	No classical Taylor series; subgradient or piecewise
Premature truncation	Order 1 on highly curved function	Systematic error (gamma ignored in finance)
Large shocks	Large or	Delta-gamma or duration-convexity underestimate risk
Conditioning	Ill-conditioned Hessian	Quadratic approximation unreliable far from