convergence of this series $a_{n}=\frac{1}{n}+\frac{1}{n+1}+....+\frac{1}{2n}$?? The sequence with nth term is given as
$a_{n}=\frac{1}{n}+\frac{1}{n+1}+....+\frac{1}{2n}$
this sequence will converge to??
what I did is:
for this we can use sandwich theorem i.e $f(x)\leq g(x)\leq h(x)$   then $lim h(x)=lim g(x)=lim f(x)$
here we can write...
$0\leq m\leq n$    
$0+n\leq m+n\leq n+n$
$\frac{1}{n}\geq \frac{1}{m+n}\geq \frac{1}{2n}$
adding this n times we get 
$1\geq\frac{1}{m+n}\geq \frac{n}{2n}$
applying lim we get 1/2..
 ..but its wrong.. how will it converge to log2??

With $h=1/n$,$$\sum_{i=n}^{2n}\frac1i=\sum_{i=n}^{2n}\frac h{ih}\to\int_{x=1}^2\frac{dx}x.$$
Alternatively, using the Harmonic numbers,
$$H_{2n}-H_{n-1}=\ln(2n)+\gamma+\frac1{4n}...-\ln(n-1)-\gamma-\frac1{2n-2}+...\to\ln2.$$
