Taylor series
A Taylor series is a power series whose coefficients are determined by a function’s derivatives evaluated at a specified expansion point.
Geometry for Programmers
A Taylor series is a power series whose coefficients are determined by the derivatives of a function at a specified center.
- How it is used
- Approximates differentiable functions with cheaper polynomial models for limited ranges.
- How deep it goes
- Explains derivative-based centers, degree selection, and SymPy generation, but does not develop convergence theory.
- Examples
- Uses exponential, logarithmic, and trigonometric expansions, including SymPy listings and a visual comparison of added terms.
- What you can do after
- Apply it via SymPy’s series function to generate finite polynomial approximations at selected centers and degrees.
Math and Architectures of Deep Learning
A multivariable Taylor expansion that approximates a function near a point and yields quadratic terms used in optimization arguments.
- How it is used
- Uses local loss expansions to justify gradient-based optimization and connect linear approximations with curvature.
- How deep it goes
- Defines one- and multidimensional expansions, explains gradient and Hessian terms, and stops short of routinely computing higher-order terms.
- Examples
- Uses an exponential expansion at zero, tangent and quadratic curve interpretations, and parameter displacements in loss landscapes.
- What you can do after
- Evaluate local loss changes and understand why negative-gradient steps provide the fastest infinitesimal descent direction.
Read how Math and Architectures of Deep Learning explains Taylor series →
Julia as a Second Language
A Taylor series is a power series representing a function as an expansion around a specified point using its derivatives.
- How it is used
- Approximates trigonometric functions by converting mathematical summations into executable Julia loops.
- How deep it goes
- Explains summation-to-loop translation and finite accuracy, but does not prove the Taylor-series definition.
- Examples
- Uses sine calculations, trigonometric tables, a Julia summation function, and comparisons with built-in sin.
- What you can do after
- Build it from scratch by implementing sine with a finite Taylor-series loop and testing its accuracy against sin.
Read how Julia as a Second Language explains Taylor series →
Related concepts
- Maclaurin seriesincludes · A Maclaurin series is a power series representing a function using its derivatives evaluated at zero.
- sine functionuses · The sine function is a periodic trigonometric function that maps a real number or angle to a value between −1 and 1.
- derivativeuses · A derivative is a mathematical quantity that measures the instantaneous rate of change of a function with respect to an input variable.
- trigonometryimplements · Trigonometry is the branch of mathematics studying relationships among angles, lengths, geometric ratios, coordinates, and periodic functions.
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