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