Geometric progression

$a, a r, a r^{2}, a r^{3}, \dots$

$T_{n} = a . r^{n - 1}$

$r = \frac{T_{n}}{T_{n - 1}}$

$S_{n} = \frac{a . (r^{n} - 1)}{r - 1}$

Geometric divergen

Example: $3 + 6 + 12 + 24 + \dots$

$\begin{aligned} | r | & \geq 1 \\ S_{\infty} & = \infty \end{aligned}$

Geometric convergen

Example: $4 + 2 + 1 + \frac{1}{2} + \dots$

$\begin{aligned} | r | & < 1 \\ S_{\infty} & = \frac{a}{1 - r} \end{aligned}$

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