A [[Series of Real Sequence|series]] of the form $a\sum_{i=0}^{\infty} r^{i}$ with $a,r \in \mathbb{R}$. # Properties By [[Partial Sums of Geometric Series]], $\sum_{n=0}^{N} r^{n}= \frac{1-r^{N+1}}{1-r}.$ By [[Convergent Geometric Series]], $\sum r^{n} = \frac{1}{1-r}$ iff $|r|<1$ and $\sum r^{n}=\infty$ if $|r|>1.$ Thus [[Radius of Convergence of Geometric Series]] is $1.$