Question

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

solve it

Answer 1

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

$Steps$

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

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

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

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

$=3\cdot \:2$

$\mathrm{Simplify}$

$=6$

Answer 2

Explanation:

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