等比数列の和

$ \begin{aligned} \sum_{k=0}^{n} ar^k &= a + ar + \cdots + ar^n \\ &= a \ \frac{1-r^{n+1}}{1-r} \ (r \neq 1) \end{aligned}

$ \begin{aligned} \lim_{n \to \infty} \sum_{k=0}^n ar^k &= \lim_{n \to \infty} a \frac{1 - r^{n+1}}{1 - r} \\ &= \left\{ \begin{aligned} & \frac{a}{1 - r} && (-1 < r < 1) \\ & -1 と 1 で振動 && (r = -1) \\ & \infty に発散 && (r \geq 1) \\ & -\infty と \infty で振動 && (r < -1) \end{aligned} \right. \end{aligned}