# Geometric Series

A $\textit{geometric series}$ is the sum of the terms of a geometric sequence.
for example:

• $1, 2, 4, 8, \cdots , 2048$ is a finite geometric sequence.
• $1+2+4+8+ \cdots +2048$ is the corresponding finite geometric series.

If we add the first $n$ terms of an infinite geometric sequence, we calculate a finite geometric series called the $n$th partial sum of the corresponding infinite series.

We have an infinite geometric series if we add all of the terms in a geometric sequence that goes on and on forever.

\begin{align} \displaystyle \require{color} S_n &= u_1 + u_2 + u_3 + \cdots + u_{n-1} + u_{n} \\ &= u_1 + u_1r + u_1r^2 + \cdots + u_1r^{n-2} + u_1r^{n-1} \\ &= \dfrac{u_1(r^n-1)}{r-1} \text{ or } \dfrac{u_1(1-r^n)}{1-r}\\ \end{align}
Ensure $r \ne 1$.

## Proof of Geometric Series Formula

\begin{align} \displaystyle \require{AMSsymbols} \require{color} S_n &= u_1 + u_1r + u_1r^2 + \cdots + u_1r^{n-2} + u_1r^{n-1} \\ rS_n &= u_1r + u_1r^2 + u_1r^3 + \cdots + u_1r^{n-1} + u_1r^{n} \\ rS_n &= (\color{red}u_1 \color{black} + u_1r + u_1r^2 + u_1r^3 + \cdots + u_1r^{n-1}) + u_1r^{n} – \color{red}u_1 \\ rS_n &= S_n + u_1r^n-u_1 \\ r S_n – S_n &= u_1r^n-u_1 \\ S_n(r-1) &= u_1(r^n-1) \\ \therefore S_n &= \dfrac{u_1(r^n-1)}{r-1} \end{align}

## Example 1

Find the sum of $3+6+12+\cdots$ to $8$ terms.

\begin{align} \displaystyle u_1 &= 3 \\ r &= 2 \\ n &= 8 \\ S_8 &= \dfrac{3(2^8-1)}{2-1} \\ &= 765 \end{align}

## Example 2

Find the sum of $0.2+0.02+0.002+\cdots$ of the first $n$ terms.

\begin{align} \displaystyle u_1 &= 0.2 \\ r &= 0.1 \\ n &= n \\ S_n &= \dfrac{0.2(0.1^n-1)}{0.1-1} \\ &= \dfrac{2(1-0.1^n)}{9} \end{align}