Question

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

Answer 1

Answer:

Answer is provided in the picture given above

Answer 2

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

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

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

$\mathrm{If\:the\:series\:is\:of\:the\:form\:}\sum _{n=0}^{\infty }r^n$

$\mathrm{If\:}|r|<1\mathrm{,\:then\:the\:geometric\:series\:converges\:to\:}\frac{1}{1-r}$

$\mathrm{If\:}|r|\ge \:1\mathrm{,\:then\:the\:geometric\:series\:diverges}$

$=\frac{1}{1-\left(\frac{1}{2}\right)}$

$\frac{1}{1-\left(\frac{1}{2}\right)}=2$

$=3\cdot \:2$

= 6