Teste da comparação do limite

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O teste da comparação do limite é um método para classificar séries quanto à convergência. Este teste é uma generalização do teste da comparação.

Teste da comparação por limite (simples)

Sejam $\sum _{n=1}^{+\infty }a_{n}$ e $\sum _{n=1}^{+\infty }b_{n}$ séries de termos positivos. Então:

Se $\lim _{n\to +\infty }{\frac {a_{n}}{b_{n}}}=C$ , sendo $C$ um número e $0<C<+\infty$ , então:

ambas as séries divergem ou ambas as séries convergem.

Se $\lim _{n\to +\infty }{\frac {a_{n}}{b_{n}}}=0$ , então:

a convergência de $\sum _{n=1}^{+\infty }b_{n}$ implica a convergência de $\sum _{n=1}^{+\infty }a_{n}$ .

Se $\lim _{n\to +\infty }{\frac {a_{n}}{b_{n}}}=+\infty$ , então:

a divergência de $\sum _{n=1}^{+\infty }b_{n}$ implica a divergência de $\sum _{n=1}^{+\infty }a_{n}$ ;

Teste da comparação por limite superior

Sejam $\sum _{n=1}^{+\infty }a_{n}$ e $\sum _{n=1}^{+\infty }b_{n}$ séries de termos positivos. Então:

Se $\limsup _{n\to +\infty }{\frac {a_{n}}{b_{n}}}<+\infty$ , temos que:

a convergência da segunda série implica a convergência da primeira.

Demonstração

É claro que basta mostrar a segunda versão mais geral do teorema.

Do limite superior temos que existe um $N$ tal que

a_{n}\leq (C+1)b_{n},~~\forall n\geq N

Aplique o teste da comparação para os somatórios a partir de N e o resultado segue.

Exemplo

Seja $b_{n}={\frac {1}{n^{3/2}}}$ e $a_{n}={\frac {\ln {n}}{n^{2}}}$ .

Como $\lim _{n\to +\infty }{\frac {a_{n}}{b_{n}}}={\frac {\ln(n)}{n^{1/2}}}=0$ , temos que:

\sum _{n=1}^{+\infty }{\frac {\ln n}{n^{2}}}

converge pois a série dos

b_{n}

é uma série harmônica generalizada que converge pelo teste da integral.

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