Prove that $\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots \cdots \cdots +\frac{1}{2n}>\frac{13}{24}$ 
Proving $$\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+\cdots \cdots \cdots +\frac{1}{2n}>\frac{13}{24}\;, n>1\;,n\in \mathbb{N}$$

$\bf{My\; Try::}$ Using Limit as a sum $$\frac{1}{n}\left[\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+\cdots \cdots \cdots +\frac{1}{1+\frac{n}{n}}\right] = \int^{1}_{0}\frac{1}{1+x}dx = \ln 2$$
But i did not understand how can i prove That $\displaystyle \bf{L.H.S}>\frac{13}{24}$
Help required, Thanks

Let $$f(n)=\frac1{n+1}+\frac1{n+2}+\cdots+\frac1{2n}=\sum_{r=1}^{2n}\frac1r-\sum_{s=1}^n\frac1s$$
So, $$f(n+1)-f(n)=\frac1{(2n+1)(2n+2)}>0$$ for integer $n\ge0$
$\displaystyle \implies f(n)$ is an increasing function.
Now, $f(2)=\frac13+\frac14=\frac7{12}>\frac{13}{24}$ as $7\cdot24>12\cdot13$
