Re: [seul-edu] [OT] summation of 1/2x

[Jan Hlavacek <jhlavacek@sf.edu>]

On Wed, Apr 05, 2000 at 04:25:00PM +0100, jm wrote:
> 
> > > I got to thinking of this last week when a friend and I were trying to
> > > remember what the summation of 1/x evaluated from x=0 to x=infinity
> > > is.
> >
> >I would like to see what you found! ;-)
> 
> If I remember well, there is a trick to calculate the sum of 1/2x from 1 to 
> infinity
> you just need a sheet of paper:
>   + first you cut in half the sheet of paper: you get 1/2 sheet and another 
> 1/2 sheet
>   + then you cut in half again one of the 1/2 sheet: you get 1/4 and 1/4
> +  then you cut in half again...
> 
> so you can prove that 1/2+1/4+1/8+..+1/2n+.. = 1

Actually, that is 1/2^n (one over 2 to the n), not 1/2n.  As a matter of
fact, the series 1/n does not have a finite sum.  But after adding some
terms at the beginning, the sum increases so slowly that from a computer
calculation, it may indeed seem like you have a convergence.  After
adding first 10000 terms, it all looks like 9.787..., and it seems that
these digits are not changing any more.  But after adding first 100000
terms you get 12.0901..., and so on.  It keeps increasing, and it is not
bounded.  Of course, when you get so far that your computer thinks that
1/n = 0, you have the "limit" :-).  Then you increase the precission,
and your "limit" suddenly changes.    

This is a classical example that demonstrates shortcommings of numerical
computations in math.  Even better example is 1/(n*ln(n)) (ln stands for
natural logarithm).  This series is also divergent, but the sum
increases even slower than the sum of 1/n.  

-- 
Jan Hlavacek                                            (219) 434-7566
Department of Chemistry                               Jhlavacek@sf.edu
University of Saint Francis               http://199.8.81.3/Jhlavacek/