\(s = SessionTime[]; 
c = \(a = 121311534567815/11621543213818718\); 

    i = 0; list = {}; 
    While[a > 0, \(i++\); 
      Print[i, "                ", Short[b = 1/Ceiling[1/a], 5], 
        "         ", SessionTime[] - s]; s = SessionTime[]; 
      a = a - 1/Ceiling[1/a]; list = Append[list, b]]\)\(Apply[Plus, list] === c
\)