I've long wondered what is the smallest n for which (*) 1 + 1/2 + 1/3 + . . . + 1/n > 100. Roughly, I thought it was where ln(n) = 100, i.e., n ~ e^100 ~ 2.688 x 10^43. But more exactly, it would be closer to where 100 - ln(n) = gamma (Euler's), i.e. let K = floor( e^(100-gamma) ). Then n should be fairly close to K (which is roughly only 1.509 x 10^43). Mathematica gives K as exactly K = 15092688622113788323693563264538101449859497 and to my surprise it claims that 1 + 1/2 + 1/3 + . . . + 1/K > 100 but 1 + 1/2 + 1/3 + . . . + 1/(K-1) < 100 (!). Can someone confirm that this K is exactly the least n for which (*) holds? Thanks, Dan P.S. Anyone know a useful upper bound U(n) for the error E(n) = |(1+1/2+1/3+...+1/n) - ln(n) - gamma| ? Evidently E(n) gets very small very fast.