Sujet : Re: hot to write out this summation:
De : invalid (at) *nospam* example.invalid (Moebius)
Groupes : sci.mathDate : 30. Jun 2024, 23:15:35
Autres entêtes
Organisation : A noiseless patient Spider
Message-ID : <v5sle7$mrbr$1@dont-email.me>
References : 1 2
User-Agent : Mozilla Thunderbird
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
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