Prove $\frac{1}{n} + \frac{1}{n+1} + .... + \frac{1}{2n} > 0.63 $ I need to prove this: 
$\frac{1}{n} + \frac{1}{n+1} + .... + \frac{1}{2n} > 0.63 $.
I know its $\log{(2)} $ and it's something like $\int_{0}^{1} \ \frac{dx}{1+x}\ =  \log{(2)}$, but I don't know the steps between those. 

Let $(a_n)$ be the sequence
$$a_n= \frac{1}{n}+\frac{1}{n+1} + \dots + \frac{1}{2n},$$
then $a_1=1.5$ and $a_{n+1}-a_n= \frac{1}{2n+2} +\frac{1}{2n+1} -\frac{1}{n}<0$, then the sequence is decreasing, as u have noticed that $a_n \to \log  2$, it won't be hard to figure out that $a_n>\log 2 \approx 0.693$ for all integers $n>0$.
