Zeno's Paradox - Walking to a Wall
One of Zeno's paradoxes involves a person attempting to walk to a wall. To reach the wall, they must:
- First go half the distance
- Then half the remaining distance
- Then half of what still remains
- And so on...
Since this process can always be continued and never ended, Zeno argued it's impossible to reach the wall. Yet we know the person can reach the wall. This paradox introduces us to infinite series.
Mathematical Representation
The total distance can be expressed as the sum of infinitely many smaller distances:
To understand this sum, we examine the partial sums \(s_n\) which represent the sum of the first n terms:
\(s_1 = \frac{1}{2} = 0.5\)
\(s_2 = \frac{1}{2} + \frac{1}{4} = 0.75\)
\(s_3 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} = 0.875\)
\(s_4 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = 0.9375\)
\(s_5 = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} = 0.96875\)
\(s_{10} \approx 0.999\)
\(s_{16} \approx 0.99998\)
\(\lim_{n \to \infty} s_n = 1\)
Interactive Partial Sums
Individual Terms:
Visualizing the Convergence
As more terms are added, the sum gets closer to 1
Watch the animation to see the convergence
As we add more terms, the partial sum approaches 1 but never reaches it. However, the sum of the infinite series equals 1, resolving Zeno's paradox and explaining how we can reach the wall in reality.