The harmonic series is the sum of all unit fractions. Every term 1/n tends to zero, which might suggest the sum converges, but it does not. The proof uses grouping: 1/3+1/4 > 1/2, then 1/5+1/6+1/7+1/8 > 1/2, and each such group adds at least 1/2, so the total exceeds any bound. Yet it diverges with extraordinary slowness: to reach a partial sum of 100 requires more terms than atoms in the observable universe.
H(n) and ln(n) grow together, always differing by approximately γ ≈ 0.5772. Both diverge: to reach H(n) = 100 requires about 10^43 terms.
~10^43 terms are needed to reach H(n)=100. More than atoms in the observable universe.
The harmonic series 1 + 1/2 + 1/3 + ... diverges, proved by Nicole Oresme around 1350. Despite every term tending to zero, the sum exceeds any bound. Partial sums grow like ln(n) + gamma where gamma ≈ 0.5772 is the Euler-Mascheroni constant. After a million terms the sum is only about 14. To reach 100 requires more than 10^43 terms. The alternating series 1 - 1/2 + 1/3 - ... converges to ln 2.