Infinito enumerável
Autor: mom el star | Publicado: 2012-06-18T12:45:00-03:00

Jamil SilvaConsidere a bije&ccedil;&atilde;o que a cada fun&ccedil;&atilde;o &fnof; : ℕ ⟶ { 0 , 1 } associa um subconjunto dos Naturais, &weierp;, se e somente se, &forall; k&isin;ℕ, &fnof;(k)= 0 &rArr; k&notin;&weierp; e &fnof;(k)= 1 &rArr; k&isin;&weierp;Assim :&nbsp;&forall;n&isin;ℕ, &fnof;(n) = 0 &rArr; &weierp; = &empty; e &fnof;(n) = 1 &rArr; &weierp; = ℕ...Ver maisCurtir&nbsp;&middot;&nbsp;&nbsp;&middot;&nbsp;Seguir (desfazer) publica&ccedil;&atilde;o&nbsp;&middot;&nbsp;h&aacute; 6 horasJamil Silva&nbsp;&fnof;(0) = 1 e &fnof;(n+1) = 0 &rArr; &weierp; = {0}&fnof;(1) = 1 e &fnof;(0) = &fnof;(n+2) = 0 &rArr; &weierp; = {1}&fnof;(a) = 1 e &fnof;(n&ne; a) = 0 &rArr; &weierp; = {a}h&aacute; 4 horas&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;Como o conjunto dos Naturais &eacute; bem ordenado, podemos dizer que cada sequencia de 0 e 1 representa um subconjunto. Assim:00000000000.... &rArr; &weierp; = &empty; = { }1000000000 .... &rArr; &weierp; = { 0 }0100000000 ... &rArr; &weierp; = { 1 }h&aacute; 4 horas&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;&lrm;1100000000 ... &rArr; &weierp; = { 0 , 1 }1010000000 ... &rArr; &weierp; = { 0, 2 }100100000 ... &rArr; &weierp; = { 0 , 3 }etc.h&aacute; 4 horas&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;Pares e &Iacute;mpares...................................................01010101010101010 .... &rArr; &weierp; = { 1 , 3 , 5 , 7 , 9 , 11, 13 , 15 , ...}1010101010101010.... &rArr; &weierp; = { 0 , 2 , 4 , 6 , 8 , 10, 12 , 14 , ...}h&aacute; 3 horas&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;&lrm;..........................................................................................................................&gt;&gt;&gt; Sabemos que cada s&eacute;rie finita de 0 ou 1 representa um n&uacute;mero natural na base dois, logo o conjunto de todas as s&eacute;ries finitas &eacute; cont&aacute;vel com cardinal igual ao dos n&uacute;meros naturais, mas e as s&eacute;ris infinitas, como cont&aacute;-las?h&aacute; 2 horas&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;&lrm;.................................................................&fnof;( k&sup2; ) = 1 e &fnof;( n&ne; k&sup2;) = 0 &rArr; 11001000010000001000000001&nbsp;....&eacute; uma s&eacute;rie infinita que diverge, mas podemos torcer para que sua representa&ccedil;&atilde;o decimal na base dois seja convergente a algum n&uacute;mero real.Ser&aacute;?Para que n&uacute;mero converge a soma:1 + 1/2 + 1/16 + 1/512 ... = ⅀{k=0 ... k =&infin;} [1/2^(- k&sup2;)] ???....................................................................h&aacute; &plusmn; 1 hora&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;parece que converge, segundo WolframAlphahttp://www.wolframalpha.com/input/?i=Sum%5B1%2F%282%5E%28k%5E2%29%5D+Sum[1/(2^(k^2)] - Wolfram|Alphawww.wolframalpha.comThe answer to Sum[1/(2^(k^2)] ... Enter what you want to calculate or know about&nbsp;in science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music...h&aacute; &plusmn; 1 hora&nbsp;&middot;&nbsp;Curtir&nbsp;&middot;&nbsp;Jamil Silva&nbsp;&lrm;.................................................................1.564468413605938579334729274272475662306258269970439046444505596028480133179578406659130640162469148447402471611685206115425419260268746661599427733814829558974562739176993844326721853328360548814611390346402771406252264355213938599101454290094331612302055686770930631733544819821861921916058895728216323664735171067922948842741927470499900476972197878067854725294657345631510058172959430874242013341800743558570458...h&aacute; &plusmn; 1 hora&nbsp;&middot;&nbsp;CurtirJamil Silva&nbsp;ERRATA.........................................................⅀{k=0 ... k =&infin;} [2^(- k&sup2;)]h&aacute; &plusmn; 1 hora&nbsp;&middot;&nbsp;Curtir