# Determine whether the geometric series is convergent or divergent. 10-4+1.6-0.64+.... If it's

Determine whether the geometric series is convergent or divergent.
$10-4+1.6-0.64+\dots .$
If it's convergent find its sum.
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Step 1
Let $S=10-4+1.60.64+\dots$
Given that S is a geometric series,
The known fact is that a geometric series $a+ar+a{r}^{2}+\dots$ is convergent $⇔|r|<1$.
By comparing the given series with its general form,
$a=10$, $r=\frac{-4}{10}=-0.4$.
$|r|=|-0.4|<1$.
This implies that the series $10-4+1.6-0.64+\dots$ is convergent.
Step 2
The known fact is that a geometric series $a+ar+a{r}^{2}+\dots$ - is equal to $\frac{a}{1-r}\left(r<1\right)$.
$10-4+1.6-0.64±\dots =\left(10+1.6+0.256+\dots \right)-\left(4+0.64+\dots \right)$

$=\left(\frac{10-4}{1-0.16}\right)$
$=\left(\frac{6}{0.84}\right)$
$=7.1428571428571429$