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From: Moebius <invalid@example.invalid>
Newsgroups: sci.math
Subject: Re: hot to write out this summation:
Date: Mon, 1 Jul 2024 00:15:35 +0200
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Am 01.07.2024 um 00:06 schrieb Moebius:
> Am 30.06.2024 um 23:38 schrieb Chris M. Thomasson:
> 
>> r[0] = .01
>> r[1] = .0011
>> r[2] = .000111
>> r[3] = .00001111
>> r[4] = .0000011111
>> ...
> 
> r[0] = .01
> r[i+1] = r[i] / 10 + 1 / 10^(2*i+2)
> 
>> Taken to infinity, it would be: .(0)(1) or something? How to properly 
>> write it:
> 
> lim_(i->oo) r[i] = 0. (zero)

Hint:

Let

s[0] = .0111111...
s[1] = .0011111...
s[2] = .0001111...
:

Then

An e IN: r[i] < s[i].

Hence:

lim_(i->oo) r[i] <= lim_(i->oo) s[i].

But cleary

       lim_(i->oo) s[i] = 0 .

(After all, s[i+1] = s[i] / 10.)

Hence: lim_(i->oo) r[i] <= lim_(i->oo) s[i] = 0.

On the other hand:

Ai e IN: 0 < r[i].

Hence 0 <= lim_(i->oo) r[i].

So we finally get:

0 <= lim_(i->oo) r[i] <= 0 ,

and hence lim_(i->oo) r[i] = 0. qed