What are the explicit formulas for the spatial and temporal sums in Zeno's paradox resolution?

Conditions

• Initial speeds $v_A = 2, v_T = 1$
• Infinite subdivision of the chase

Reasoning, step by step

1. Identify the cumulative spatial requirement as the sum of all intervals Achilles traverses.
2. Write the series for space: $2 + 1 + 1/2 + 1/4 + \dots$
3. Identify the sequence of temporal durations required to traverse each interval.
4. Write the series for time: $1 + 1/2 + 1/4 + 1/8 + \dots$

Example

The animation aggregates the endless steps into two summations displayed at the bottom: $s_{\text{total}} = 2 + 1 + \frac{1}{2} + \frac{1}{4} + \cdots$ and $t_{\text{total}} = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots$.

Common misconceptions

• Assuming the time series starts with 2 like the distance series; it starts with 1 because the first interval takes 1 unit of time.
• Thinking the series diverges; they are explicitly described as convergent.

Watch the explanation

Explore next

• What are the finite limits of these two series?
• How do these series relate to the relative speed calculation?