Question

Determine se as séries são convergentes ou divergentes

a) $\displaystyle \sum_{n=1}^{\infty}\frac{2}{n^{0,85}}$

b) $\displaystyle 1+\frac{1}{8} +\frac{1}{27} +\frac{1}{64} +\frac{1}{125}+....$

c) $\displaystyle 1+\frac{1}{3} +\frac{1}{5} +\frac{1}{7} +\frac{1}{9}+....$

Answer 1

a)
 $\sum_{n=1}^{\infty}\frac{2}{n^{0,85}} = 2\sum_{n=1}^{\infty}\frac{1}{n^{0,85}} \to \text{p serie com } p\ \textless \ 1 \\\\\\ \sum_{n=1}^{\infty}\frac{2}{n^{0,85}} \; , diverge$

b) 
a1 = 1
a2 = 1/8 = 1/2³
a3 = 1/27 = 1/3³ 
an = 1/n³

$\sum_{n=1}^{\infty}\frac{1}{n^{3}} \to \text{p serie com } p\ \textgreater \ 1 \\\\ \sum_{n=1}^{\infty}\frac{1}{n^{3}} , converge$


c)

observando o denominador
a1 = 1
a2 = 3 = 2*2 - 1
a3 = 5 = 2*3 -1
a4 =7 = 2*4 -1
an = 2*n -1

$\sum_{n=1}^{\infty}\frac{1}{2n -1}$

temos que:
$\frac{1}{2n-1} \ \textless \ \frac{1}{2n}$

como:
$\sum_{n=1}^{\infty}\frac{1}{2n} = \frac{1}{2} \sum_{n=1}^{\infty}\frac{1}{n} \to \text{serie harmonica , divergente}$


pelo teste da comparação
$\sum_{n=1}^{\infty}\frac{1}{2n -1} , \; \text{diverge}$