Geometric Series

Definition: Geometric Series

A geometric series is a real series ${\sum }_{n=1}^{\infty }{a}_n$ such that the sequence $(a_n{)}_{n\in \mathbb{N}}$ can be specified as

$$a_n=q^{n-1}a$$

for two real numbers $a$ and $q$ with $q\ne 0$.

Theorem: Product of Members of Geometric Series

If $a_p$ and $a_m$ are members of a geometric series, then for all $k\in \mathbb{N}$ with $k<m$, we have

PROOF

TODO

Theorem: Partial Sums of Geometric Series

The $n$-th partial sum $S_n$ of a geometric series $a_n$ with ratio $q$ is given by

$$S_n=\frac{1-q^n}{1-q}{a}_1$$

PROOF

TODO

Theorem: Convergence of Geometric Series

A geometric series $(a_n{)}_{n\in \mathbb{N}}$ with ratio $q$ converges absolutely if and only if $|q|<1$ and diverges. Furthermore, if $(a_n{)}_{n\in \mathbb{N}}$ converges, then

$$\sum _{n=1}^{\infty }{a}_n=\frac{a_1}{1-q}$$

PROOF

TODO