Question

$\sum _{n=0}^{\infty }\frac{3}{2^n}solve it$

solve

Answer 1

$\mathrm{Check\:convergence\:of\:}\sum _{n=0}^{\infty \:}\frac{3}{2^n}:\quad 6$

$\mathrm{Apply\:the\:constant\:multiplication\:rule}:\quad \sum c\cdot a_n=c\cdot \sum a_n$

$=3\cdot \sum _{n=0}^{\infty \:}\frac{1}{2^n}$

$\mathrm{Apply\:Series\:Geometric\:Test}:\quad 2$

$=3\cdot \:2$

$\mathrm{Simplify}$

$=6$

Answer 2

Explanation:

$\sf \to \: \sum _{n=0}^{\infty }\frac{3}{2^n} \\ \\ \sf \to \: \sum _{n=0}^{\infty }n = 3 \times 2 \\ \\ \sf \to \red{\sum _{n=0}^{\infty }n \: = 6}$