Verifique se as sequências abaixo são convergentes ou divergentes. Justifique suas respostas.
(cn), onde
, para todo
.
(cn), onde
Passo 1
Beleza, vamos lá! O enunciado nos pede para verificar se a sequência
, que está limitada por
e
para todo
natural, é convergente ou divergente. Para resolver isso, vamos precisar entender sobre limites de sequências e a Regra do Aperto (também conhecida como Teorema do Confronto).
Passo 2
Primeiro, vamos analisar os limites das sequências que estão confinando
. Temos duas sequências:
-
: Sabemos que . Conforme cresce, cresce exponencialmente, então tende a 0. -
: Essa é uma sequência do tipo onde . À medida que aumenta, também cresce muito, fazendo com que tenda a 0.
Passo 3
Agora que sabemos que ambas as sequências
e
tendem a 0 quando
vai para o infinito, podemos usar a Regra do Aperto. Essa regra diz que se uma sequência
está sempre entre duas outras sequências que têm o mesmo limite
, então
também tende a
.
No nosso caso:
E vimos que:
Passo 4
Aplicando a Regra do Aperto:
Como está sempre entre duas sequências que tendem a 0,
também deve tender a 0. Isso significa que a sequência
é convergente e seu limite é 0.
Como
Resposta
A sequência
é convergente e seu limite é 0.
Exercícios relacionados
<html><head></head><body><div>Verifique se as sequências abaixo são convergentes ou divergentes. Justifique suas respostas.<br>
<span class="math-inline ">
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</body></html>
<span class="math-inline ">
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<html><head></head><body><div>Verifique se as sequências abaixo são convergentes ou divergentes. Justifique suas respostas.<br>
<span class="math-inline ">
<mjx-container class=“MathJax” jax=“SVG”><svg style=“vertical-align: -0.912ex;” xmlns=“http://www.w3.org/2000/svg” width=“15.191ex” height=“2.878ex” role=“img” focusable=“false” viewBox=“0 -868.9 6714.3 1271.9”><title>a_n = (n + 1) / (3n - 3)</title><g stroke=“currentColor” fill=“currentColor” stroke-width=“0” transform=“scale(1,-1)”><g data-mml-node=“math”><g data-mml-node=“mo”><path data-c=“28” d=“M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z”></path></g><g data-mml-node=“mi” transform=“translate(389,0)”><path data-c=“1D44E” d=“M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 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<span class="math-inline ">
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Estude a convergência das séries:
b) ;
b)
Estude a convergência das séries:
а) ;
а)