Re: Sum of Infinite Series

["Paul O. Johnson" <pojhome@SWBELL.NET>]

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!