Re: Where do I begin?

On Fri, 29 Apr 2005 20:07:47 -0600, Virgil
<ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote:

>In article <q2p571l72a4d9khbper39mi03i63u439vu@xxxxxxx>,
> David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
>
>> On Fri, 29 Apr 2005 13:22:31 -0600, Virgil
>> <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote:
>>
>> >In article <19f471dgjcdaod4nr1ebb5frruogpvop4t@xxxxxxx>,
>> > David C. Ullrich <ullrich@xxxxxxxxxxxxxxxx> wrote:
>> >
>> >> On Fri, 29 Apr 2005 04:46:54 -0700, William Elliot
>> >> <marsh@xxxxxxxxxxxxxxxxxx> wrote:
>> >>
>> >> >On Fri, 29 Apr 2005, David C. Ullrich wrote:
>> >> >> This is not as easy as it looks at first. The answer does not follow
>> >> >> from the standard "alternating series test", because the terms do
>> >> >> not decrease in absolute value.
>> >> >
>> >> >If (aj)_j decreases to a limit of zero,
>> >> >then sum(j=1,..) (-1)^j aj converges ?
>> >> >
>> >> >Do it take much to prove that?
>> >>
>> >> That's not hard to prove - it's in any decent calculus
>> >> book. But it's not relevant here, because the a_j do
>> >> not decrease.
>> >>
>> >>
>> >> ************************
>> >>
>> >> David C. Ullrich
>> >
>> >They do, but as you remarked elsewhere, not monotonically, which is a
>> >part of the requirement.
>>
>> Huh??? What does it mean for a sequence to decrease, but not
>> monotonically?
>
>Perhaps I expressed that badly.
>
>In an alternating series, the terms may go to zero without each term
>being no larger in absolute value than its predecessor.
>
>For example a_n, with a_{2*n}= 1/2^n and a_{2*n-1} = -1/3^n.
>The terms "decrease in absolute value towards zero, but not
>monotonically" in the sense I intended.

Doubtless that's so. But if we're talking about what the
words _mean_ as opposed to what you intended for them to
mean, then no, those terms do _not_ decrease in absolute
value.

Think about it this way: If the phrase "decrease to zero"
means what you evidently intended for it to mean then
_any_ sequence of positive numbers that tends to zero
would be said to "decrease to zero" - with this definition
of the word "decrease" the term means nothing at all.

>
>> (Except for worrying about whether we're talking
>> about weak or strict monotonicity, which doesn't matter here,
>> a sequence is monotone precisely when it's either increasing
>> or decreasing.)
>>
>> ************************
>>
>> David C. Ullrich


************************

David C. Ullrich
.

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