The Infinite Staircase

Yes — the insect does reach the far end. The harmonic series guarantees it by eventually summing past any finite target. The time required is approximately e^100,000 seconds — finite but incomprehensibly large.

The key quantity is not the insect's absolute position but its fractional position along the rope. The rope stretches uniformly, so the insect's fraction of total rope covered each second is what matters. Step 1: At second 1: rope is 100,000 cm. Insect walks 1 cm → covers 1/100,000 of the rope. After stretching, rope becomes 200,000 cm, but the insect's fractional position is preserved. Step 2: At second 2: rope is 200,000 cm. Insect walks 1 cm → covers 1/200,000 additional fraction of the rope. Step 3: At second n: rope is n × 100,000 cm. Insect walks 1 cm → covers 1/(n × 100,000) additional fraction. Step 4: Total fractional distance covered after N seconds = (1/100,000) × (1 + 1/2 + 1/3 + ... + 1/N) = H(N) / 100,000. Step 5: H(N) is the harmonic series. It diverges — it grows without bound as N increases, just very slowly. Step 6: The insect reaches the end when H(N) = 100,000. Since H(N) ≈ ln(N), we need N ≈ e^100,000 — an unimaginably large but mathematically finite number. Key Insight: The harmonic series 1 + 1/2 + 1/3 + ... diverges — it has no finite upper bound. This single mathematical fact guarantees the insect eventually completes its journey, despite the stretching appearing catastrophic at every step.