Convergence or Divergence of a Series college algebra This is super elementary, so I apologize in advanced:
$$\frac{4}{3} - \frac{8}{9} + \frac{16}{27} - \frac{32}{81} + \ldots$$
Does this series converge or diverge?
I am trying to find a pattern but neither taking out $4/3$ or $4$ seems to work any suggestions would be greatly appreciated. I am sorry again for being so elementary. I am not in this class just trying to refresh myself with algebra for tutoring

The series is converging.
As you can see, the terms are given by $x_n=(-1)^{n-1}\frac{2^{n+1}}{3^n}=(-1)^{n-1}\cdot2\cdot (\frac{2}{3})^n$ where $n\geq1$
Now, we have
$|x_{n+1}|-|x_n|=2(\frac{2}{3})^{n+1}-2(\frac{2}{3})^n<0$
$\lim_{n\to\infty} |x_n|=\lim_{n\to\infty}2\cdot (\frac{2}{3})^n=0$ since $\frac{2}{3}<1$
Thus, by the alternating series test, the series converges.
For the sake of completion, put $\sum x_n=\sum_{n=1}^\infty -2\cdot(\frac{-2}{3})^n =-2\sum_{n=1}^\infty (\frac{-2}{3})^n=-2(-\frac{2}{5})=\frac{4}{5}$
QED.
