{"id":160,"date":"2019-05-06T13:39:00","date_gmt":"2019-05-06T13:39:00","guid":{"rendered":"http:\/\/localhost\/?p=160"},"modified":"2019-05-06T14:06:20","modified_gmt":"2019-05-06T14:06:20","slug":"exercicios-series","status":"publish","type":"post","link":"http:\/\/localhost\/index.php\/2019\/05\/06\/exercicios-series\/","title":{"rendered":"Exerc\u00edcios (S\u00e9ries)"},"content":{"rendered":"

$\\newcommand{\\N}{\\mathbb N}$ $\\newcommand{\\Z}{\\mathbb Z}$ $\\newcommand{\\Q}{\\mathbb Q}$ $\\newcommand{\\R}{\\mathbb R}$<\/p>\n

1. Sejam $\\sum a_n$ e $\\sum b_n$ s\u00e9ries convergentes tais que $\\sum a_n=a$ e $\\sum b_n=b$. Demonstre as seguintes:<\/p>\n

    \n
  1. $\\sum(a_n + b_n)=a+b$;<\/li>\n
  2. $\\sum(\\alpha a_n)=\\alpha a$;<\/li>\n
  3. $\\sum (a_nb_n)$ n\u00e3o \u00e9 necessariamente igual a $ab$.<\/li>\n<\/ol>\n

    2. Define para $x\\in\\R$ as seguintes fun\u00e7\u00f5es.<\/p>\n

      \n
    1. $e(x)=\\sum_{n\\geq 0}x^n\/n!$;<\/li>\n
    2. $s(x)=\\sum_{n\\geq 0}(-1)^n x^{2n+1}\/(2n+1)!$;<\/li>\n
    3. $c(x)=\\sum_{n\\geq 0}(-1)^n x^{2n}\/(2n)!$;<\/li>\n
    4. $sh(x)=\\sum_{n\\geq 0}x^{2n+1}\/(2n+1)!$;<\/li>\n
    5. $ch(x)=\\sum_{n\\geq 0}x^{2n}\/(2n)!$.<\/li>\n<\/ol>\n

      Demonstre que essas fun\u00e7\u00f5es s\u00e3o bem definidas no sentido que as s\u00e9ries s\u00e3o convergentes para todo $x\\in\\R$. Demonstre que as s\u00e9ries s\u00e3o absolutamente convergentes.<\/p>\n

      3. Decide quais das seguintes s\u00e9ries s\u00e3o absolutamente convergentes, condicionalmente convergentes, ou divergentes.<\/p>\n

        \n
      1. $\\sum_{n\\geq 1} 1\/n$;<\/li>\n
      2. $\\sum_{n\\geq 1} 1\/n^2$;<\/li>\n
      3. $\\sum_{n\\geq 4} 1\/(n-3)$;<\/li>\n
      4. $\\sum_{n\\geq 1} 1\/(n^2+2)$;<\/li>\n
      5. $\\sum_{n\\geq 1} 1\/3^n$;<\/li>\n
      6. $\\sum_{n\\geq 1} 1\/(3^n+1)$;<\/li>\n
      7. $\\sum_{n\\geq 1} 1\/\\ln n$;<\/li>\n
      8. $\\sum_{n\\geq 1} n^2\/(n^4+2)$;<\/li>\n
      9. $\\sum_{n\\geq 1} n\\cdot\\mbox{sen}^2\\,n\/(n^3+1)$;<\/li>\n
      10. $\\sum_{n\\geq 1} 1\/(n!)$;<\/li>\n
      11. $\\sum_{n\\geq 1} 1\/((2n)!)$;<\/li>\n
      12. $\\sum_{n\\geq 1} n!\/((2n)!)$;<\/li>\n
      13. $\\sum_{n\\geq 1} (2n)!\/(n!(n+1)!)$;<\/li>\n
      14. $\\sum_{n\\geq 1} 1\/(r^n n!)$ com $r>1$;<\/li>\n
      15. $\\sum_{n\\geq 1} (-1)^n\/(2^n)$;<\/li>\n
      16. $\\sum_{n\\geq 1} (-1)^n\/(2n)$;<\/li>\n
      17. $\\sum_{n\\geq 1} (-1)^n(1+1\/n^2)$;<\/li>\n
      18. $\\sum_{n\\geq 1} (-1)^n\/(n^4+7)$.<\/li>\n<\/ol>\n

        4. Seja $\\sum_{n\\geq 0}a_n$ uma s\u00e9rie condicionalmente convergente e defina $$ p_n=\\left\\{\\begin{array}{ll} a_n & \\mbox{se $a_n\\geq 0$}\\\\ 0 & \\mbox{se $a_n<0$}\\end{array}\\right. $$ e $$ q_n=\\left\\{\\begin{array}{ll} -a_n & \\mbox{se $a_n< 0$}\\\\ 0 & \\mbox{se $a_n\\geq 0$}\\end{array}\\right. $$ Demonstre que as s\u00e9ries $\\sum p_n$ e $\\sum q_n$ s\u00e3o divergentes.<\/p>\n

        5. Seja $\\sum_{n\\geq 0}a_n$ uma s\u00e9rie condicionalmente convergente e seja $c\\in\\{\\infty,-\\infty\\}$. Esbo\u00e7e uma reordena\u00e7\u00e3o $\\sum a_{\\varphi(n)}$ da s\u00e9rie $\\sum a_n$ tal que $\\sum a_{\\varphi(n)}=c$.<\/p>\n","protected":false},"excerpt":{"rendered":"

        $\\newcommand{\\N}{\\mathbb N}$ $\\newcommand{\\Z}{\\mathbb Z}$ $\\newcommand{\\Q}{\\mathbb Q}$ $\\newcommand{\\R}{\\mathbb R}$ 1. Sejam $\\sum a_n$ e $\\sum b_n$ s\u00e9ries convergentes tais que $\\sum a_n=a$ e $\\sum b_n=b$. Demonstre as seguintes: $\\sum(a_n + b_n)=a+b$; $\\sum(\\alpha a_n)=\\alpha a$; $\\sum (a_nb_n)$ n\u00e3o \u00e9 necessariamente igual a $ab$. 2. Define para $x\\in\\R$ as seguintes fun\u00e7\u00f5es. $e(x)=\\sum_{n\\geq 0}x^n\/n!$; $s(x)=\\sum_{n\\geq 0}(-1)^n x^{2n+1}\/(2n+1)!$; $c(x)=\\sum_{n\\geq 0}(-1)^n … Continue reading Exerc\u00edcios (S\u00e9ries)<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[1],"tags":[],"_links":{"self":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/160"}],"collection":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/comments?post=160"}],"version-history":[{"count":4,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/160\/revisions"}],"predecessor-version":[{"id":167,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/posts\/160\/revisions\/167"}],"wp:attachment":[{"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/media?parent=160"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/categories?post=160"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/index.php\/wp-json\/wp\/v2\/tags?post=160"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}