Useful mathematical tools

1. $\ln n$ $\le$ $\sum\limits_{k = 1}^{n} \frac{1}{k}$ $\le$ $\ln n$ $+$ $1$ for $n$ $\in$ $\mathbb{Z}^{+}$
2. $\sum\limits_{k = 1}^{n}ar^{k-1}$ $=$ $a\frac{r^{n} - 1}{r - 1}$ if $r$ $\neq$ $1$, $n$ $\in$ $\mathbb{Z}$