P.G soma

por **Victor Gabriel** » Dom Abr 14, 2013 21:01

Determine o valor da soma: S = 0,2+0,22+0,222+..
Qual é o valor 3S?

assim: S= $S=\frac{0,2}{1-10/11}$
S=2,2
logo S.3= 2,2.3=6,6
Estou certo?

por **e8group** » Seg Abr 15, 2013 21:08

Acredito que está incorreto .

Observe que ,

$\underbrace{0,22}_{a_2} = \underbrace{0,2}_{a_1} + \underbrace{0,2}_{a_1} \frac{1}{10} = \underbrace{0,2}_{a_1} (1 + \frac{1}{10})$

$\underbrace{0,222}_{a_3} = \underbrace{0,2}_{a_1} + \underbrace{0,22}_{a_2} \frac{1}{10} = \underbrace{0,2}_{a_1} + (\underbrace{0,2}_{a_1} + \underbrace{0,2}_{a_1} \frac{1}{10}) \frac{1}{10} = \underbrace{0,2}_{a_1} (1 + \frac{1}{10} +\frac{1}{100}$

$\underbrace{0,2222}_{a_4} = \underbrace{0,2}_{a_1} + \underbrace{0,222}_{a_3} \cdot \frac{1}{10} = \underbrace{0,2}_{a_1} + \underbrace{0,2}_{a_1} (1 + \frac{1}{10} +\frac{1}{100} ) \cdot \frac{1}{10} = \underbrace{0,2}_{a_1}(1 + \frac{1}{10} +\frac{1}{100} + \frac{1}{1000} )$

(...)

$\underbrace{0,22\hdots22}_{a_n} = \underbrace{0,2}_{a_1}(1 + \frac{1}{10}+ \frac{1}{10^2} + \hdots + \frac{1}{10^{n-1}} )$

(...)

Imagine a soma destas infinitas parcelas $a_1 + a_2+ \hdots + a_n + \hdots = 0,2 + 0,2(1 +\frac{1}{10}) + 0,2(1 +\frac{1}{10} +\frac{1}{100} ) + \hdots + 0,2(1 +\frac{1}{10} +\frac{1}{100} + \hdots + \frac{1}{10^{n-1}} ) + \hdots > 0.2 + 0.2 + 0.2 + \hdots$ .

Qual a sua estimativa para o valor da soma acima ?

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