Answer
By dividing any latter term by its former we can get the common ration ($r$). $ 0.5\div 2 = 0.25$ or $\frac{1}{4}$
$r$ = |$\frac{1}{4}|\lt1 $ which means that this geometric series will converge.
Now, use the formula from definition 4 in page 750 , knowing that $a$ refers to the first term which is $2$ and using the calculated $r$ and we get
$S= \frac{a}{1-r}$ = $\frac{2}{1-\frac{1}{4}} = \frac{2}{\frac{3}{4}} = \frac{8}{3}$
Result: the series converges and its sum is $\frac{8}{3}$ .
Work Step by Step
We look at the ratio change from term to term and we find that:
By dividing any latter term by its former we can get the common ration ($r$). $ 0.5\div 2 = 0.25$ or $\frac{1}{4}$ . Also, $0.125\div 0.5 = 0.25$ .
Note that |$\frac{1}{4}|\lt1 $ which means that this geometric series will converge.
Now, use the formula from definition 4 in page 750 , knowing that $a$ refers to the first term which is $2$ and using the calculated $r$ and we get
$S= \frac{a}{1-r}$ = $\frac{2}{1-\frac{1}{4}} = \frac{2}{\frac{3}{4}} = \frac{8}{3}$
Result: the series converges and its sum is $\frac{8}{3}$ .
