Question

Qual é o valor da soma:
S=$\frac{1}{2} -\frac{1}{4} + \frac{1}{8} - \frac{1}{16} +... ?$

Answer 1

Explicação passo-a-passo:

Temos duas progressões geométricas infinitas de razão $\sf \dfrac{1}{4}$

• $\sf PG_1\Big(\dfrac{1}{2},~\dfrac{1}{8},~\dfrac{1}{32},~\dfrac{1}{128},~\dots\Big)$

• $\sf PG_2\Big(-\dfrac{1}{4},-\dfrac{1}{16},-\dfrac{1}{64},\dots\Big)$

A soma dos termos de uma PG infinita é dada por:

$\sf S=\dfrac{a_1}{1-q}$

=> Primeira PG

$\sf S=\dfrac{\frac{1}{2}}{1-\frac{1}{4}}$

$\sf S=\dfrac{\frac{1}{2}}{\frac{4-1}{4}}$

$\sf S=\dfrac{\frac{1}{2}}{\frac{3}{4}}$

$\sf S=\dfrac{1}{2}\cdot\dfrac{4}{3}$

$\sf S=\dfrac{4}{6}$

$\sf S=\dfrac{2}{3}$

=> Segunda PG

$\sf S=\dfrac{-\frac{1}{4}}{1-\frac{1}{4}}$

$\sf S=\dfrac{-\frac{1}{4}}{\frac{4-1}{4}}$

$\sf S=\dfrac{-\frac{1}{4}}{\frac{3}{4}}$

$\sf S=-\dfrac{1}{4}\cdot\dfrac{4}{3}$

$\sf S=-\dfrac{4}{12}$

$\sf S=-\dfrac{1}{3}$

Logo:

$\sf S=\dfrac{2}{3}-\dfrac{1}{3}$

$\sf S=\dfrac{2-1}{3}$

$\sf \red{S=\dfrac{1}{3}}$