Answer
Converges
Work Step by Step
Consider $u_n=(-1)^{n+1}(0.1)^n$
Here, $u_n$ is positive for all the values of $n$.
and $|u_n|=(0.1)^n$
Thus, the series $\Sigma_{n=1}^\infty |u_n| $ is a convergent geometric series with $r=0.1$
Hence, the series converges absolutely.