Harmonic Series (mathematics) - Rate of Divergence

Rate of Divergence

The harmonic series diverges very slowly. For example, the sum of the first 1043 terms is less than 100. This is because the partial sums of the series have logarithmic growth. In particular,

where is the Euler–Mascheroni constant and ~ which approaches 0 as goes to infinity. This result is due to Leonhard Euler. He proved also the more striking fact that the sum which includes only the reciprocals of primes also diverges, i.e.