Geometric Progression
\(a, \: ar, \: ar^2, \: ar^3, \: \dotso \)
\(T_n = a \:.\: r^{n-1}\)
\( r = \dfrac{T_n}{T_{n - 1}} \)
\(S_n = \dfrac{a \:.\: (r^n - 1)}{r - 1}\)
Sum to Infinity
Geometric divergen
Example: \(3 + 6 + 12 + 24 + \dotso\)
\begin{equation*} \begin{split} | r | & \geq 1 \\\\ S_{\infty} & = \infty \end{split} \end{equation*}
Geometric convergen
Example: \(4 + 2 + 1 + \frac 12 + \dotso\)
\begin{equation*} \begin{split} | r | & < 1 \\\\ S_{\infty} & = \frac{a}{1 - r} \end{split} \end{equation*}
Exercise