The Most Counterintuitive Sum in Math

The harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges to infinity. Cross out every term whose denominator contains the digit 9, and the series converges to approximately 22.92. In 1914, Aubrey Kempner proved this with a three-page paper and a geometric bound of 80. It took until 1979 for Robert Baillie to compute the actual value: 22.92067 66192 64150 34816... The convergence is so slow that after 10^24 terms, you're still more than 1 away from the answer. The same trick works for any digit. Removing digit 1 gives ~16.18. Removing digit 0 gives ~23.10. The ordering follows Benford's law. And in 2008, Bakir Farhi proved that for any digit, the "exactly k copies" variants all converge to the same universal limit: 10 ln 10 ≈ 23.026. The pattern extends to multi-digit strings: removing "42" gives ~228, removing "314159" gives ~2.3 million, always following the formula 10^k ln 10. We can compute this number to thousands of digits, but we cannot prove it irrational, and no closed form is known. ⏱ CHAPTERS 0:00 The Series 1:08 Why It Diverges 2:50 The Decimation 5:08 Kempner's Bound 7:09 The Real Answer 9:01 Every Digit 11:08 The Limit 13:02 Open 📚 FURTHER READING • OEIS A082838 — The Kempner constant (sum excluding 9) • Kempner, "A Curious Convergent Series" (1914), AMM Vol. 21 • Baillie, "Sums of Reciprocals of Integers Missing a Given Digit" (1979), AMM Vol. 86 • Schmelzer & Baillie, "Summing a Curious, Slowly Convergent Series" (2008), AMM Vol. 115 • Farhi, "A curious result related to Kempner's series" (2008) • Irwin, "A curious convergent series" (1916), AMM Vol. 23 • Wikipedia: Kempner series 🎵 MUSIC • "Origami" by Scott Buckley — https://www.youtube.com/watch?v=y30VTqITSw4 • "Echoes Of Home" by Scott Buckley — https://www.youtube.com/watch?v=MsajW80n-2o • "Incredulity" by Scott Buckley — https://www.youtube.com/watch?v=aciVgU_hJoc #kempnerseries #harmonicseries #mathematics #convergence #numbertheory #mathvideo