How many members of the harmonic series 1+1/2+1/3+1/4+ …+1/n are needed to add up close to 10 , without going over it?

(In reply to A lower bound and a stab at a solution by Jer)

Indeed, the 12367th term is 0.0000808603541683512573785073178620522357887927549122665..., which causes the total to pass 10:
 

            
thru term    total             
 12365       9.9998812810285453761196033775747563846278551079819424280...
 12366       9.9999621479216393434493785676119551554510800796785298451...
 12367       10.000043008275807694706757074929817207686868872433442111...
 

But, of course, after 12365 terms I'd still think that the total of over 9.99988 is near 10, though not as near as possible.

Term 7501, which is 0.000133315557925609918677509665377949606719..., is the one that brings the sum nearer to 10 than to 9: 9.50007394516904546697686643134742082130442987....
 
    5   point 15
   10   for I=1 to 100000
   15       Pprev=Prev:Prev=T
   20       T=T+1/I
   25       if T>=9.5 and Prev<9.5 then print I:print 1/I:print Prev:print T
   30       if T>=10 then print I:print 1/I:print Pprev:print Prev:print T:cancel for:goto 1000
  100   next
 1000   end

Edited on November 1, 2011, 12:34 pm

Posted by Charlie on 2011-11-01 12:34:22