Problem Statement

In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: The series of harmonic numbers thus obtained is often loosely referred to as the harmonic series. Harmonic numbers are closely related to the Riemann zeta function, and roughly approximate the natural logarithm function; differing by γ (lowercase Gamma), the Euler–Mascheroni constant. The harmonic series is divergent, albeit quite slowly, and grows toward infinity.

Let's start with the solution:

  [ $ "bigrat.qky" loadfile ] now!
 
  0 n->v
  20 times 
    [ i^ 1+ n->v 1/v v+
      2dup 20 point$ echo$
      say " = " 
      2dup vulgar$ echo$ cr ] 
  2drop
  cr
  1 temp put
  0 n->v 1
  [ dup dip
      [ n->v 1/v v+
       temp share n->v 2over v< ]
    swap if 
      [ temp share echo
        say " : "
        dup echo cr 
        1 temp tally ]
    temp share 11 < while
    1+ 
    again ]
  temp release
  drop 2drop