Geometric Series

Description

This animation visualizes the infinite geometric series 1 + 1/2 + 1/4 + 1/8 + ... by repeatedly halving a unit square. Each term is shown as a filled rectangle that fits beside the previous ones like puzzle pieces, and the partial sums are displayed numerically, all converging toward the limit of 2.

---

Phases

#	Phase Name	Duration	Description
1	Introduction	5s	Title + formula S = a/(1-r) shown; parameters a=1, r=1/2 displayed
2	First square	4s	Full unit square (1×1) appears; label "1"
3	Halving steps	20s	6 iterations: each new rectangle (1/2, 1/4, ...) appears beside; running sum label updates
4	Convergence callout	8s	Arrow/limit annotation shows sum → 2; formula evaluated S=1/(1-0.5)=2
5	Summary	7s	All visible; convergence condition

---

Layout

+-----------------------------------------------+
|  S = a/(1-r)   a=1, r=1/2, S=2  [top formula] |
|                                                |
|  +----+--+-+--+                  [squares]    |
|  | 1  |1/2|1/4|...                            |
|  +----+--+-+--+                               |
|                                                |
|  Sum = 1 + 1/2 + 1/4 + ... → 2   [bottom]    |
+-----------------------------------------------+

Area Descriptions

Area	Content	Notes
Top	Formula + parameters	MathTex, updates as r is labeled
Center	Rectangle mosaic	Unit square then halved rectangles side by side
Running total	Partial sum decimal	Updates each step: 1, 1.5, 1.75, ...
Bottom	Limit formula + arrow	Shows convergence toward 2

---

Assets & Dependencies

• Fonts: LaTeX
• Colors: Colors cycle through BLUE, GREEN, YELLOW, ORANGE, RED, PURPLE for each term; BG = #1a1a2e
• Manim version: ManimCE 0.19.1

---

Notes

• Squares are placed left-to-right; each new piece is half the width of the previous
• All rectangles share the same height (say 2 Manim units) and are anchored at the bottom-center
• The "2 units wide" target boundary drawn as a dashed vertical line