Newsgroups: sci.math
Subject: Re: 1+1/2+1/3+1/4+...+1/n
From: jsv@math.canterbury.ac.nz (Julian Visch)
Date: Fri, 20 Nov 1992 00:03:46 GMT
Organization: Department of Mathematics, University of Canterbury

In article <HANCHE.92Nov19123151@ptolemy.ams.sunysb.edu>, hanche@ams.sunysb.edu (Harald Hanche-Olsen) writes:
|> >>>>> On Thursday, 19 Nov 1992 13:23:29 CET, Mutter Christoph Johannes
|> >>>>> <K3032E2@ALIJKU11.BITNET> said:
|> 
|> Chris> Hallo.
|> Chris> I've a problem. I have to calculate the sum 1+1/2+1/3+1/4+...+1/n.
|> Chris> The result should be 100. But my computer is far too slow, to solve this
|> Chris> problem. I need the index n when the sum > 100.0
|> Chris> And that exactly.
|> Chris> Has anyone calculated this? (Perhaps on a CRAY|) ?
|> 
|> You won't find your answer by brute force on any present-day computer.
|> Consider the fact that this sum is fairly close to log(n)+gamma for
|> large n, where gamma is Euler's constant; it follows that the desired
|> n is rather close to exp(100-gamma), which has the approximate value
|> 15092688622113788323693563264538101449859497.364099... (according to
|> Maple).  Some finesse (asymptotic analysis) is clearly called for to
|> get the exact answer.  Hint: Estimate the area between the step
|> function 1/[t] (where [t] is the largest integer <= t) and 1/t for t
|> between n and infinity.
|> 
|> - Harald

Using Maple with 1000 decimal places I found that for
n = 15092688622113788323693563264538101449859496 the sum was just below 100

i.e 
> evalf(sum(1./n,n=1..15092688622113788323693563264538101449859496));
99.999999999999999999999999999999999999999999942747074071711130782941148738689\
110197701561042496005255810450666769067029302578130750912902932624408446045107\
869792495620899224319418245033025025182041993034298490182584171857999337302532\
300688073637257930075532467393740472811084063270156466843499470742536200190655\
917162237352085261170317735188104551661757782435748221008019753912713374271191\
245994582381205887879299026740996025832728915343973670530181721654188180444513\
866671269077935205401265297418980406002802771943272916252123760413277867268520\
425498261036541860051418778270807265148213766262785646305055826160117200136416\
808713860003120853843066723275631027241605891026022366052237104221394063066151\
733625632162542850742304726146277045601427472123573240266841661206726172029358\
724453487706763774112615731347877669277984229681395187863668103744366396464141\
100727348414962089305298508586435048307651285171855167950037159251939422672229\
76667807374084860658417534406817092726803488494861982494218688745

while for n = 15092688622113788323693563264538101449859497 the sum was just above

i.e 

> evalf(sum(1./n,n=1..15092688622113788323693563264538101449859497));
100.00000000000000000000000000000000000000000000900432120889789165585816073577\
751939961860129721706615382739050887641575408610184470055411778370615584870751\
680997954613077090825707297214593275638141168220303234871249026721287533644787\
097357219223720460156707881723358750375106238732796886967352576151981740578473\
154443538850098393109855711492559264156512012581065894522985030576856661670380\
492197666026527806563072034551189051406419969136807866826955384937185305480516\
267185059582831185433249645216295298417893230548338419330257569719612514260620\
553579091533904193838602944051540368095054244965838316856354047269852341672224\
205459789283269124232309456277027340683218367802976813905201876499321839890129\
646653916571531782513179351436863392002769576844850881159970997259946269145700\
283456130069508391884565985862432025625268231022831416269193315035632732401558\
707000407239207005124432307481794262410739446246664908789671874005330617916534\
62604840757764909792941256264546443643288565468699021237948437324

Newsgroups: sci.math