Prove inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$

Question

Prove the inequality $1 < \frac{1}{n} + \frac{1}{n+1} + \ldots + \frac{1}{3n-1} < 2$ For all $n \in \mathbb{N}$

I've done the right hand side, but can't do the left side of the inequality. For the right:

$\frac{1}{n} + \ldots + \frac{1}{3n-1} < \underbrace{\frac{1}{n} + \ldots + \frac{1}{n}}_{2n} = 2$

Now I haven't been able to make progress with the left.

Answer 1

Hint:

Show $\dfrac{1}{n} \lt \dfrac{1}{n+k}+\dfrac{1}{3n-1-k} \lt \dfrac{2}{n} $ for $0 \le k \lt n$

Then sum over $k$