Lista III

Cálculo E

Date: 27-4-99

Exercício 1   Teste a convergência ou divergência das séries:

1.

$\sum\limits_{n=1}^\infty \frac{\sqrt{n}}{n^2+1}$

2.

$\sum\limits_{n=1}^\infty \frac{10^n}{n!}$

3.

$\sum\limits_{n=1}^\infty \frac{(-1)^n \ln n}{\sqrt{n}}$

4.

$\sum\limits_{n=1}^\infty \frac{\tan \frac{1}{n}}{n}$

5.

$\sum\limits_{n=1}^\infty \frac{e^{1/n}}{n^2}$

6.

$\sum\limits_{n=1}^\infty (\sqrt[n]{2}-1)^n$

7.

$\sum\limits_{n=1}^\infty \frac{(-1)^n\sqrt{n}}{n+5}$

8.

$\sum\limits_{n=2}^\infty \frac{1}{(\ln n)^{\ln n}}$

Exercício 2   Estude os limites:

1.

$\lim\limits_{n\to \infty} \frac{n+3}{n^3+4}$.

2.

$\lim\limits_{n\to \infty} \frac{n!}{n^n}.$

3.

$\lim\limits_{n\to \infty} \sqrt[n]{a}$

4.

$\lim\limits_{n\to \infty} \sqrt[8]{n^2+1}-\sqrt[4]{n+1}$.

5.

$\lim\limits_{n\to \infty} \sqrt[n]{n^2+n}.$