Question

Sabendo que a série geométrica ...

Answer 1

Usando o que foi dado no enunciado e as propriedades de somas e séries, temos que

$\displaystyle\sum_{n=1}^{\infty}\left(\dfrac{4}{n(n+1)}+\dfrac{1}{2^n} \right )\\\\\\ =\sum_{n=1}^{\infty}\dfrac{4}{n(n+1)}+\sum_{n=1}^{\infty}\dfrac{1}{2^n}\\\\\\ =4\cdot \sum_{n=1}^{\infty}\dfrac{1}{n(n+1)}+\sum_{n=1}^{\infty}\left(\dfrac{1}{2} \right )^{\!\!n}\\\\\\ =4\cdot \sum_{n=1}^{\infty}\dfrac{1}{n(n+1)}+\sum_{n=1}^{\infty}\dfrac{1}{2}\cdot\left(\dfrac{1}{2} \right )^{\!\!n-1}\\\\\\ =4\cdot 1+\dfrac{\frac{1}{2}}{1-\frac{1}{2}}\\\\\\ =4+\dfrac{\left(\frac{1}{2}\right)}{\left(\frac{1}{2}\right)}\\\\\\ =4+1\\\\ =5$


Bons estudos! :-)