Mostre que a sequência
definida por
é convergente.
Passo 1
Beleza, vamos dar uma olhada no que a questão está pedindo. A gente tem uma sequência definida por:
O objetivo aqui é mostrar que essa sequência é convergente, ou seja, que ela se aproxima de um valor específico conforme
cresce. Para isso, vamos precisar entender o que é uma sequência convergente e aplicar alguns conceitos de limites.
Passo 2
Primeiro, vamos revisar o que significa uma sequência ser convergente. Uma sequência
é convergente se existe um número
tal que, conforme
fica cada vez maior, os termos
ficam cada vez mais próximos de
. Em outras palavras:
Então, nosso objetivo é encontrar esse
ou, pelo menos, mostrar que ele existe.
Passo 3
Agora, vamos analisar a sequência dada:
Perceba que cada termo seguinte adiciona uma fração que envolve o fatorial de
. Sabemos que o fatorial
cresce muito rápido conforme
aumenta, o que faz com que
fique cada vez menor.
Passo 4
Para mostrar que a sequência é convergente, vamos considerar o que acontece quando
tende ao infinito. Vamos somar todos esses termos infinitamente:
Essa é uma série conhecida na matemática chamada série exponencial, e sabemos que ela converge para o número
, que é aproximadamente 2,71828.
Passo 5
Como a série infinita
converge para
, significa que a sequência das suas somas parciais
também converge para
. Ou seja:
Resposta
A sequência
é convergente e seu limite é o número
.
Exercícios relacionados
Sejam
tal que
e
uma sequência cujo termo geral é definido por
Mostre que
é monótona e limitada, conclua a partir daí que existe
. Além disso mostre que
Mostre que
não existe.
a)Suponha que a sequência
converge para zero. Encontre
.
b)Calcule
b)Calcule
<html><head></head><body><div>Mostre que a sequência <span class="math-inline ">
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<div><span class="math-block ">
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<div>é divergente.</div>
</body></html>
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<div><span class="math-block ">
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<div>é divergente.</div>
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