Sequences and Series

Zeno's Paradox

Infinitely many steps = finite distance

  • High school
  • about 3 min
  • 10 of 10 in the family
Not started
1/21/41/81/16S=0.98438→ 1
Steps: 6
∑n=1∞\ rac12n=\ rac12+\ rac14+\ rac18+⋯=1\sum_{n=1}^{\infty}\ rac{1}{2^n} = \ rac{1}{2}+\ rac{1}{4}+\ rac{1}{8}+\cdots = 1
S6=1−126=0.984375S_{6} = 1 - \frac{1}{2^{6}} = 0.984375

Zeno's paradox: "Achilles must traverse infinitely many steps." But an infinite sum of finite numbers can be finite! 1/2+1/4+1/8+... equals exactly 1.