Taylor Series Approximation

Description

Demonstrates how the Taylor series builds increasingly accurate polynomial approximations of cos(x) around x=0 by successively adding terms. Each new term is animated in a new color, revealing how the approximation extends further from the center.

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Phases

#	Phase Name	Duration	Description
1	Setup	5s	Title, axes, plot cos(x) in white
2	T₀ = 1	6s	Plot constant term, explain 0th-order approximation
3	T₂ = 1 - x²/2	7s	Add quadratic term, new color, better fit near origin
4	T₄ = 1 - x²/2 + x⁴/24	7s	Add quartic term, fits further out
5	T₆ = add -x⁶/720	7s	Add sixth-degree term, excellent fit over wider range
6	T₈	6s	Add 8th-degree term, near-perfect over [-π, π]
7	Formula display	8s	Show general Taylor series formula, error term O(x^n)
8	Summary	4s	All approximations overlaid, convergence message

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Layout

+--------------------------------------------------+
|  Title: "Taylor Series Approximation"            |
+--------------------------------------------------+
|                                                  |
|   Axes with cos(x)           | Formula panel    |
|   (white curve)              | (right side)     |
|                              |                  |
|   T₀ (red)                   | cos(x) = Σ       |
|   T₂ (orange)                | (-1)^n x^(2n)    |
|   T₄ (yellow)                | ─────────────    |
|   T₆ (green)                 |    (2n)!         |
|   T₈ (teal)                  |                  |
|                              | Current term     |
|                              | highlighted      |
+--------------------------------------------------+

Area Descriptions

• Left 60%: Axes from -2π to 2π showing cos(x) and successive Taylor polynomials
• Right 40%: Taylor series formula with current partial sum highlighted
• Bottom: Degree label and approximation quality description

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Assets & Dependencies

• Fonts: LaTeX
• Manim version: ManimCE 0.19.1

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Notes

• cos(x) Taylor series centered at 0: Σ (-1)^n x^(2n)/(2n)!
• Terms: 1, -x²/2!, x⁴/4!, -x⁶/6!, x⁸/8!
• Colors cycle: RED → ORANGE → YELLOW → GREEN → TEAL
• Use axes x_range = [-2π, 2π], y_range = [-1.5, 1.5]
• Clip polynomial plots to avoid large values far from origin