I was fooling around with this today (avoiding fixing
the oven, I'm sure I'll have to pay for that soon),
and I found an interesting twist on this series.

I wasn't (not having noticed or been around for the
previous round that Paul mentioned) quite sure
how the series for 1/3 might have been derived,
so I was trying to come up with a demonstration
of it, hoping I could find a general formula.

I had this gut feeling that the series Myke listed had
popped out of an Euler transformation somehow, and I was
fooling around with that. What I noticed, however, was that
if the Euler transformation was applied backward of what
I had first thought (and I'm not this smart, this is an
immediate consequence of a derivaton in R.W. Hamming's
book on Numerical Methods, section 12.6, on page 203 of
the Dover edition), that you can improve the convergence
of such a series while still working exclusively with
powers of two. In particular, the series

  1/2 * ( 1/4 + 1/16 + 1/64 + 1/256 + ... + 1/(4^n) + ... )

Also converges to 1/3, but at a much quicker rate.
I calculated the two series in Excel to demonstrate:

Index   Myke's Series   Powers of 1/4
1       0.5             0.25
2       0.25            0.3125
3       0.375           0.328125
4       0.3125          0.33203125
5       0.34375         0.333007813
6       0.328125        0.333251953
7       0.3359375       0.333312988
8       0.33203125      0.333328247
9       0.333984375     0.333332062
10      0.333007813     0.333333015
11      0.333496094     0.333333254
12      0.333251953     0.333333313
13      0.333374023     0.333333328
14      0.333312988     0.333333332
15      0.333343506     0.333333333
16      0.333328247     0.333333333
17      0.333335876     0.333333333
18      0.333332062     0.333333333
19      0.333333969     0.333333333
20      0.333333015     0.333333333
21      0.333333492     0.333333333
22      0.333333254     0.333333333
23      0.333333373     0.333333333
24      0.333333313     0.333333333
25      0.333333343     0.333333333
26      0.333333328     0.333333333
27      0.333333336     0.333333333
28      0.333333332     0.333333333
29      0.333333334     0.333333333
30      0.333333333     0.333333333
31      0.333333333     0.333333333

Thus, the replacement series requires almost exactly half
as many terms to achieve the same degree of accuracy.

Not that this answers Myke's question, but I
thought it was kind of useful.

I suppose I really should try to fix the oven
before my wife starts getting cranky.

--Bob

On Sat, Jul 24, 1999 at 09:36:03AM -0400, Myke Predko wrote:
> Hi Folks,
>
> Quick question for the Math Wizzes out there.
>
> Is there a series method of doing a divide by a constant?  I'm thinking of
> the divide by three operation:
>
> n/3 = x/2 - x/4 + x/8 - x/16 + x/32 - x/64...
>
> which is very easy to code and is very fast.  Of course, powers of two work
> the same way.
>
> Is there a general case for non-powers of two?
>
> I got asked for a routine which needs a divide by seven and wondered if
> there was a better way of doing it instead of the methods which use a
> variable.
>
> Thanx,
>
> myke

--
============================================================
Bob Drzyzgula                             It's not a problem
bob@drzyzgula.org                until something bad happens
============================================================