# Determine whether the geometric series is convergent or divergent. If it is conv

Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
$10-2+0.4-0.008+\dots$
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Geometric series:
$\sum _{n=0}^{\mathrm{\infty }}a\cdot {r}^{n}=\frac{a}{1-r}$
Where r and a are constants
If |r|<1, then the series converges to $\frac{a}{1-r}$
$10-2+0.4-0.08\dots$
$10+10\cdot \left(-\frac{1}{5}\right)+10\cdot {\left(-\frac{1}{5}\right)}^{2}+10\cdot {\left(-\frac{1}{5}\right)}^{3}\dots$
$\sum _{n=0}^{\mathrm{\infty }}10\cdot {\left(-\frac{1}{5}\right)}^{n}$
This is a geometric series with common ration $r=-\frac{1}{5}$ and Initial Term $a=10$
Since $|r|=\frac{1}{5}<1$, the given geometric series converges.
Sum of the geometric series is
$S=\frac{a}{1-r}=\frac{10}{1-\left(-\frac{1}{5}\right)}=\frac{10}{1+\frac{1}{5}}=\frac{10}{\frac{6}{5}}=\frac{50}{6}=\frac{25}{3}$
The series converges to $\frac{25}{3}$