Question

$\sum _{n=0}^{\infty }\frac{3}{2^n}$ me ajudeeem

Answer 1

$\mathrm{Remover\:a\:constante}:\quad \sum c\cdot a_n=c\cdot \sum a_n$$=3\cdot \sum \:_{n=0}^{\infty \:}\frac{1}{2^n}$$[tex]r=\frac{1}{2},\:|r|=0.5\ \textless \ 1,\:\mathrm{pelo\:teste\:de\:series\:geometricas}$$=3\cdot \frac{1}{1-\frac{1}{2}}$$\mathrm{Simplificar\:}3\cdot \frac{1}{1-\frac{1}{2}}:\quad 6$$=6$
Série geométrica:
$\mathrm{Se\:a\:serie\:possui\:a\:forma\:}\sum _{n=0}^{\infty }r^n$$\mathrm{Se\:}|r|\ \textless \ 1\mathrm{,\:entao\:a\:serie\:geometrica\:converge\:para\:}\frac{1}{1-r}$$\mathrm{Se\:}|r|\ge \:1\mathrm{,\:entao\:a\:serie\:e\:divergente}$