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Archimedes' Infinite Sum

Archimedes' Infinite Sum
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2019-06-28 04:40
SequenceCalculusLimitGeometric ConstructionGraphing
This example demonstrates Archimedes' geometric approach to approximating the area of a circle using infinite series. Key math concepts include series summation, limits, continuity, upper and lower sums (Riemann sums), piecewise functions, geometric transformations, and geometric distribution. It is constructed using GeoGebra commands like Point, Intersect, Sequence, Polygon, and Text, with a slider to control the number of terms in the series, visually showing the approximation process. Suitable for high school students and math enthusiasts studying calculus and geometric concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Intersect(xAxis, yAxis) → B
  3. Sequence(((2^(k) - 1) / 2^(k), (2^(k) - 1) / 2^(k)), k, 0, n_1 - 1, 1) → l1
  4. Polygon(B, A, 4) → poly1, f, g, h, i, C, D
  5. Sequence((1, y(l1(k))), k, 1, n_1, 1) → l2
  6. Sequence(((2^(k) - 1) / 2^(k), (2^(k - 1) - 1) / 2^(k - 1)), k, 1, n_1, 1) → l4
  7. Sequence(Polygon[l1(k), l4(k), 4], k, 1, n_1 - 1, 1) → l5
  8. Sequence(Polygon[l1(k), l2(k), 4], k, 1, n_1 - 1, 1) → l3
  9. Sequence(Polygon[l4(k), l2(k), 4], k, 1, n_1 - 1, 1) → l6
  10. Sequence(Polygon[l1(k) + (0, 1 / 2^(k)), l4(k) + (0, 1 / 2^(k)), 4], k, 1, n_1 - 1, 1) → l7
  11. Text(Sum[Sequence[FractionText[l5(k)] + (If[k < n_1 - 1, "+", ""]) + (If[k ≟ 8, "+...", ""]), k, 1, n_1 - 1]], (1.03, 0.2), true, true) → text2
  12. Text(Sum[Sequence[FractionText[l5(k)] + (If[k < n_1 - 1, "+", ""]) + (If[k ≟ 8, "+...", ""]), k, 1, n_1 - 1]], (1.03, 0.35), true, true) → text1
  13. Text(Sum[Sequence[FractionText[l5(k)] + (If[k < n_1 - 1, "+", ""]) + (If[k ≟ 8, "+...", ""]), k, 1, n_1 - 1]], (1.03, 0.5), true, true) → text3