User:Theta4/sandbox

Delta Definition
$$\Delta x = x_1 - x_0$$

Infinite Series - Easy
$$\sum_{i=1}^{\infty} \frac{1}{2^i}$$

$$= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots$$

$$= 1$$

Infinite Series - Not So Easy
$$\sum_{i=0}^{\infty} \left(-\frac{1}{2}\right)^{i}$$

$$= 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \frac{1}{32} + \ldots$$

$$= \left(1 + \frac{1}{4} + \frac{1}{16} + \ldots\right) - \left(\frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \ldots\right)$$

$$= \left[1 + \frac{1}{2} + \frac{1}{4} + \ldots - \left(\frac{1}{2} + \frac{1}{8} + \ldots\right)\right] - \left(\frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \ldots\right)$$

$$= 2 - 2 \left(\frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \ldots\right)$$

$$= 2 - \left(1 + \frac{1}{4} + \frac{1}{16} + \ldots\right)$$

$$= 2 - \sum_{i=0}^{\infty} \left(\frac{1}{4}\right)^i$$

$$= 2 - \frac{1}{1 - \frac{1}{4}}$$

$$= 2 - \frac{1}{\frac{3}{4}}$$

$$= 2 - \frac{4}{3} = \frac{2}{3}$$