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.
Geometric Series

Geometric Series Formula

If we are adding the first $n$ terms of an infinite geometric sequence, we are then calculating a finite geometric series called the $n$th partial sum of the corresponding infinite series.

If we are adding all of the terms in a geometric sequence which goes on and on forever, we have an infinite geometric series.

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

Proof of Geometric Series Formula

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

Example 1

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

Example 2

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