inequality - Prove that $frac{1}{n+1} + frac{1}{n+3}+cdots+frac{1}{3n-1}>frac{1}{2}$

Without using Mathematical Induction, prove that

$$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$

I am unable to solve this problem and don't know where to start. Please help me to solve this problem using the laws of inequality. It is a problem of Inequality.

Edit: $n$ is a positive integer such that $n>1$.

Answer

The sum can be written as

$$\begin{align} \frac{1}{n+1} + \frac{1}{n+3} + \ldots + \frac{1}{3n - 1} & = \sum_{i=1}^n \frac{1}{n + 2i - 1}. \end{align}$$
Now recall the AM-HM inequality:

$$\frac 1n\sum_{i=1}^n(n + 2i - 1) > \frac{n}{\sum_{i=1}^n \frac{1}{n + 2i - 1}}.$$
(The requirement that $n > 1$ guarantees that the inequality is strict.)

Rearrange to get

$$\begin{align} \sum_{i=1}^n \frac{1}{n + 2i - 1} & > \frac{n^2}{\sum_{i=1}^n(n + 2i - 1)} = \frac 12. \end{align}$$