Bob Drzyzgula <pic@DRZYZGULA.ORG> wrote:

>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 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

...  all of which amounts to multiplying the Numerator (N) by the binary
expansion of 1/the Denominator (1/D). Note that the binary digits of 1/3
are a repeating fraction, 0.01010101......, and that the positions with
1's correspond to the terms that are present in Bob's series. Take the
same expansion of any fraction, and do likewise. In Myle's original case,
1/7 = 0.0010010010......

Dave