The Limit of a Sequence

The Original Paradox

The story goes like this: Achilles (a swift Greek hero) races against a tortoise. Since Achilles runs much faster than the tortoise, the tortoise is given a head start. Zeno then presents the following argument:

1. Achilles must first reach the point where the tortoise started (point A).
2. By the time Achilles reaches point A, the tortoise has moved to a new position (point B).
3. Achilles must then reach point B.
4. But by the time he reaches point B, the tortoise has moved to point C.
5. This process continues infinitely - whenever Achilles reaches the tortoise's previous position, the tortoise has moved ahead.
6. Therefore, Achilles can never overtake the tortoise.

Mathematical Interpretation

Mathematically, this paradox involves an infinite series. In our example:

• The tortoise starts 35% of the track ahead of Achilles (at 40% vs Achilles at 5%)
• Achilles runs 5 times faster than the tortoise

Then the sequence of distances Achilles must cover would be:

• First, Achilles runs 35% of the track to reach the tortoise's starting point
• By then, the tortoise has moved 7% (35%/5), so Achilles must run 7% more
• By then, the tortoise has moved 1.4% (7%/5), so Achilles must run 1.4% more
• By then, the tortoise has moved 0.28% (1.4%/5), and so on...

This creates the infinite series: 35 + 7 + 1.4 + 0.28 + ... = 35 × (1 + 1/5 + 1/25 + 1/125 + ...)

Resolution Through Limits

The paradox is resolved through the concept of limits. The infinite series above is a geometric series with first term a = 35 and common ratio r = 1/5.

The sum of such a series when |r| < 1 is given by:

This means Achilles will travel an additional 43.75% beyond his starting point (5%), reaching the 48.75% mark on the track where he catches the tortoise. Despite requiring an infinite number of steps in Zeno's argument, the actual distance and time are finite.

Our Visualization

In our visualization:

1. Initial Setup: Achilles starts at the 5% mark, and the tortoise has a head start at the 40% mark (Point A).
2. The Race Simulation: When you click "Start Race," Achilles first moves to the tortoise's initial position. Meanwhile, the tortoise moves forward but at a slower rate. This process repeats, simulating the sequence of positions.
3. Convergence to a Limit: The positions of both Achilles and the tortoise form sequences that converge to a specific point on the track. At this limit point (48.75%, marked as the "Meeting point" in red), Achilles catches up to and overtakes the tortoise.
4. Mathematical Resolution: The visualization demonstrates that even though we can divide the race into an infinite sequence of movements, the sum of time (or distance) for these movements is finite.

Connection to Limits

This paradox is particularly relevant to understanding limits because it shows how infinite processes can yield finite results, which is the essence of calculus and the study of limits.

The key insight is that an infinite sum of decreasing terms can converge to a finite value, just like how the sequence \(\{a_n\} = \{1/n\}\) converges to 0 as n approaches infinity.