(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 20111101 12:34:22 