Having an immense amount of trouble trying to prove the general case for inequalities involving fractions. For instance the proof is :Prove for all natural numbers,

$(\frac{1}{2^n+1}+\frac{1}{2^n+2}+\frac{1}{2^n+3}+\ldots+\frac{1}{2^{n+1}})>\frac{1}{2}$

I understand the pattern after a little help but I do not know how to finish the proof.

I am having difficulty writing a summation for the general pattern of repeating fractions and therefore using correct notation to prove the result. How should I go about doing this? 

For $n=1$

$\frac{1}{3}+\frac{1}{4}>\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$

For $n=2$

$\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}>\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}=\frac{1}{2}$