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• Proof without words: Knopp series for (pi)

  451-452

  Ángel Plaza

  Views:

  30

  There are many expressions for number π as infinite series or infinite product (see for example [1, 2, 3]). In [1] the following series for number π is attributed to K. Knopp:...

• Proof without words: four circles

  307-309

  Ángel Plaza

  Views:

  10

  Theorem. Let O, P and Q be three points on a line, with P lying between O and Q. Semicircles are drawn on the same side of the line with diameters OP, PQ and OQ. An arbelos is the figure bounded by these three semicircles. Draw the perpendicular…

• Proof without words: limit of a recursive sequence

  121-122

  Pilar Abreu

  Esther Guerra

  Ángel Plaza

  Views:

  2

  x_0 > 0 …

• Proof without words: partial sum and sum of a geometric series

  423

  Ángel Plaza

  Views:

  4

  Let r be a positive real number such that 0 < r < 1, then:…

• Proof without words

  207

  Ángel Plaza

  Views:

  6

  | cos α + sin α | ≤ √2 with equality…