Can $\sum(-1)^{n}$ be thought of as a geometric series? Although we know that $\sum(-1)^{n}$ diverges by the definition of Alternating Series. Can we say that it also diverges by using Geometric Series test?
In this situation the $|R|$ value is not $<1$ so thus the series diverges?
Is this possible?
Yes, you can think of the alternating series $\sum(-1)^{n}$ as a geometric series $\sum R^{n}$ with ratio $R=-1$.
Yes, it diverges because the ratio $R$ does not satisfy $|R|<1$.
You can also look at the partial sums: $$ \sum_{k=0}^n R^k = \frac{1-R^{n+1}}{1-R} = \frac{1-(-1)^{n+1}}{2} $$ This sequence does not converge because it oscillates between $0$ and $1$.