Useful maths equations

Geometric series

\[ {\sum^{\infty }_{n = 0} r^n} = \frac 1 {1 - r} \quad \text{if } |r| < 1 \]

\[ {\sum^{\infty }_{n = 1} r^n} = \frac {r} {1 - r} \quad \text{if } |r| < 1 \]

\[ \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + ... = \frac 1 2 {\sum^{\infty }_{n = 0} \frac{1}{2^n}} = \frac 1 2 \frac 1 {1-\frac 1 2} = 1 \]

integration by parts

\[ \int u \mathrm{d} v = uv - \int v \mathrm{d} u \]