References: Taylor series

Source material

Source curriculum (structural mirror, cited as further study):
This Clawdemy lesson bundles two consecutive 3Blue1Brown chapters into one
capstone, because both develop the single capability of approximating a function
by a polynomial built from its derivatives:
• 3Blue1Brown, Essence of Calculus, Chapter 14: "Taylor series"
  https://www.3blue1brown.com/lessons/taylor-series
• 3Blue1Brown, Essence of Calculus, Chapter 15: "Taylor series (geometric view)"
  https://www.3blue1brown.com/lessons/taylor-series-geometric-view
  Creator: Grant Sanderson
  Series index: https://www.3blue1brown.com/?topic=calculus
  License: copyright Grant Sanderson; videos published on his site and YouTube
Clawdemy's lessons are original prose that follows the pedagogical arc of this
series. We do not reproduce or transcribe the videos; we cite them as the
recommended companion. All rights to the original videos remain with the creator.

Watch this next

• Taylor series (3Blue1Brown) by Grant Sanderson. The chapter this lesson mirrors, and one of the most celebrated in the series. Watching the polynomial gain terms and wrap ever more tightly around a curve makes the matching property something you see. Paired with Taylor series (geometric view), which draws out the term-by-term geometric meaning. Together they are the natural finale to the series, as this lesson is to the track.

Going deeper

• Essence of Calculus (full series) by 3Blue1Brown. The complete series this track has followed. Rewatching the first chapter after finishing here closes the same loop the track does: the circle-area question that opened everything, answered, with the full machinery now in hand.

• Khan Academy: Calculus for a slower, exercise-driven pass over Taylor and Maclaurin series, with practice problems and immediate feedback.

Adjacent topics

Where this track has taken you, and where it points.

• The whole of Track 8. Taylor is the synthesis: it needs the power rule (the polynomial structure), the trig and exponential derivatives (the canonical series), the chain rule (composing series), higher-order derivatives (the ingredients), and the limit (convergence). The small-angle approximation and L’Hopital’s rule, met earlier as intuitions, are revealed here as Taylor’s first-order term.

• Optimization in the AI tracks. Gradient descent is a first-order Taylor step on the loss; Newton’s method is second-order; the neural tangent kernel is a first-order Taylor expansion of a network at initialization. With Track 8 complete, the matrix manipulations and the optimization arguments in a machine-learning paper read as moves you understand rather than opaque symbols, which was the point of the track from the first lesson.