Re: Sum of Infinite Series

[Brian Whatcott <inet@INTELLISYS.NET>]

Paul,
I greatly respect and admire your willingness to call a response into question
for further inspection.   I sometimes wonder who it is who writes notes to this
 list over my name. Here is my best interpretation of the writer's intention.

If I write a decimal  fraction like this
0.99999.... and ask you the limit, you will probably say, on sight: 1

If I write a binary fraction like this, and ask the same question:
 0.111111....   you will likely conclude the limit is also 1.
That's the way of fractions in any base, for that matter.
People who were in computing long ago, became familiar with
binary, octal and hexadecimal.
The "translation" of a binary fraction 0.1111...
is  one-half plus one-quarter plus one-eighth plus one-sixteenth ......

If I match a series  1/2  + 1/4 + 1/8 ... and a series
    1/6 + 1/12 + 1/24....I can easily see that each term below is
one third the size of the corresponding term above, so
I treat the limits of the series by the same factor: limit one and limit 1/3

Now, if I massage a series 1/2 + 1/4 + 1/6 + 1/8 ....
to the form of 1 + 1/3 + 1/5 + 1/7.....

I can always match each term in the second series against a smaller
term in the first series. Specifically, I can see that the first term of the
second series is twice as big as the first term in the original series.

That implies that using the same process, I can create a third series
whose first term is twice as big at least, as the first term of the second
series.   This is evidence for a series which increases without limit.

Now, to deal with my prejudice: I tried to stack books in an oblique stack
of constant step, and this limited quickly, unless I translated and rotated
each book. In this case I could continue the helical stack through two
editions of Enc Brit.
I hope I have made this amplification slightly less muddy

Brian W

At 09:16 AM 1/26/02, you wrote:
> Brian,
>
> Your answer to my question reads like those of my teachers who loved to say
> "It is obvious that . . . " before they wrote a solution on the board that
> was not at all obvious to anyone.
>
> Why is your method of  breaking my series into several more solvable series
> for computer types? Why is the limit of 1/2 + 1/4 + 1/8 + 1/16 . . . READILY
> given as 1? Why is the limit of 1/6 + 1/12 + 1/24 + . . . VISIBLY equal to
> 1/3? Why is the limit of 1/10 + 1/20 + 1/40 + . . . SIMILARLY given as 1/5?
> Why does the sum of the series 1 + 1/3 + 1/5 + 1/7 + . . . increase without
> limit?
>
> I'm sorry for my mathematical density, but none of your statements are
> obvious to me.
>
> Paul O. Johnson
>
> ----- Original Message -----
> From: "Brian Whatcott" <inet@INTELLISYS.NET>
> To: <PHYS-L@lists.nau.edu>
> Sent: Thursday, January 24, 2002 7:51 PM
> Subject: Re: Sum of Infinite Series
>
>
> > At 05:26 PM 1/24/02, you wrote:
> > ...The top step (between blocks 1 and 2) is 1/2 block-length, the next
> step
> > down (between blocks 2 and 3) is 1/4 block-length, then 1/6, 1/8, 1/10,
> > 1/12, and finally 1/14 block-length between blocks 7 and 8.
> >
> > >I want to state in the exhibit's sign what the absolute maximum extension
> > >is. This requires that I sum the infinite series 1/2 + 1/4 + 1/6 + 1/8 +
> . . .
> > >Paul O. Johnson
> >
> >
> > For computer types, the limit of the expression 1/2 + 1/4 + 1/8 + 1/16...
> > is readily given as one.
> > This leaves the series 1/6 + 1/10 + 1/12 + 1/14  + 1/18 + 1/20 ....
> >
> > Taking the series 1/6 + 1/12 + 1/24... from it,
> > this is visibly 1/3 of the first series, and limits at 1/3
> >
> > Taking the series 1/10 + 1/20 + 1/40 from it, this limits similarly at 1/5
> > Leaving us finally with a sum of 1 + 1/3 + 1/5 + 1/7 ....
> > which increases without limit as I recall, agreeing with my prejudice
> >   that there is an offset stack which can grow without limit.
> > (I thought, with a constant width of overlap)
> > Brian Whatcott
> >    Altus OK                      Eureka!
> >

Brian Whatcott
  Altus OK                      Eureka!