Question

Determine a soma dos elementos da seguinte PG: 4/10 + 4/100 + 4/1000 + ...

Answer 1

Dada a seguinte P.G.,

$\left(\dfrac{4}{10},\;\dfrac{4}{100},\;\dfrac{4}{1\,000},\;\ldots,\;\dfrac{4}{10^n},\;\ldots \right )\,,~~~~n\in\mathbb{N}^{*}$


queremos saber o valor da soma infinita de seus termos:

$S_{\infty}=\dfrac{4}{10}+\dfrac{4}{100}+\dfrac{4}{1\,000}+\ldots+\dfrac{4}{10^n}+\ldots\\\\\\ S_{\infty}=\displaystyle\sum_{k=1}^{\infty}\dfrac{4}{10^k}\\\\\\ S_{\infty}=4\sum_{k=1}^{\infty}\dfrac{1}{10^k}\\\\\\ S_{\infty}=4\sum_{k=1}^{\infty}\left(\dfrac{1}{10} \right )^{\!\!k}$
que é uma série geométrica de razão $q=\frac{1}{10}.$


Como $|q|< 1,$ a serie converge para

$S_{\infty}=\dfrac{a_1}{1-q}\\\\\\ S_{\infty}=\dfrac{\frac{4}{10}}{1-\frac{1}{10}}\\\\\\ S_{\infty}=\dfrac{\frac{4}{10}}{\frac{10}{10}-\frac{1}{10}}\\\\\\ S_{\infty}=\dfrac{4}{10-1}\\\\\\ \boxed{\begin{array}{c}S_{\infty}=\dfrac{4}{9} \end{array}}$