Question

Qual é a soma dos termos desta série??

Answer 1

Olá!

Lembre da convergência da série geométrica:

$\displaystyle \sum_{n=0}^{\infty}r^n=\dfrac{1}{1-r},\;\;\text{se}\;\;|r|<1.$

Daí,

$\displaystyle \sum_{n=0}^{\infty}\dfrac{3^n}{\pi^{n+1}}=\sum_{n=0}^{\infty}\dfrac{3^n}{\pi^n\cdot \pi}=\dfrac{1}{\pi}\sum_{n=0}^{\infty}\left(\dfrac{3}{\pi}\right)^n=\dfrac{1}{\pi}\left(\dfrac{1}{1-\frac{3}{\pi}}\right)=\\ \\ \\ = \dfrac{1}{\pi}\left(\dfrac{1}{\frac{\pi-3}{\pi}}\right)=\dfrac{1}{\pi}\cdot\dfrac{\pi}{\pi-3}=\dfrac{1}{\pi-3}.$

Bons estudos!