Question

calcule a soma dos termos da pg 1/3 + 1/9 + 1/27 infinita

Answer 1

$P.G.\ \left(\dfrac{1}{3};\ \dfrac{1}{9};\ \dfrac{1}{27};\ \cdots\right)\\\\\\q=\dfrac{a_n}{a_{n-1}},\text{ para }n\in\mathbb{N}\text{ e }n\geq2\\\\\\q=\dfrac{a_2}{a_{2-1}}\longrightarrow{q}=\dfrac{a_2}{a_1}\longrightarrow{q}=\dfrac{\dfrac{1}{9}}{\dfrac{1}{3}}\longrightarrow{q}=\dfrac{1}{9}\cdot\dfrac{3}{1}\longrightarrow{q}=\dfrac{1}{3}\\\\\\\\\lim_{n\to\infty}S_n=\dfrac{a_1}{1-q},\text{ para }-1\ \textless \ q\ \textless \ 1\\\\\\\lim_{n\to\infty}S_n=\dfrac{\dfrac{1}{3}}{1-\dfrac{1}{3}}\longrightarrow\\\\\\\lim_{n\to\infty}S_n=\dfrac{\dfrac{1}{3}}{\dfrac{2}{3}}\longrightarrow\lim_{n\to\infty}S_n=\dfrac{1}{3}\cdot\dfrac{3}{2}\longrightarrow\\\\\\\boxed{\lim_{n\to\infty}S_n=\dfrac{1}{2}}$