[tex] \sum\limits_{k=1}^{\infty}12\left(0.7\right)^{k-1} [/tex]
The infinite geometric series is converges if |r| < 1.
We have r = 0.7 < 1, therefore our infinite geometric series is converges.
The sum S of an infinite geometric series with |r| < 1 is given by the formula :
[tex] S=\dfrac{a_1}{1-r} [/tex]
We have:
[tex]a_1=12(0.7)^{1-1}=12(0.7)^0=12\\\\r=0.7[/tex]
Substitute:
[tex]S=\dfrac{12}{1-0.7}=\dfrac{12}{0.3}=40[/tex]
Answer: c. Converges, 40.
