Prove by induction that for any $n \ge 2$ : $\sum_{k=n+1}^{2n}\frac{1}{k} \ge \frac{13}{24}$

Question

Prove by induction that for any $n \ge 2$ : $$\sum_{k=n+1}^{2n}\frac{1}{k} \ge \frac{13}{24}$$

I obviously know the base case is true, but beyond that it seems to get convoluted and I'm not sure how to solve this question.

Anyone help?

Answer 1

Hint: You just need to show \begin{eqnarray*} \frac{1}{n+1} < \frac{1}{2n+1} + \frac{1}{2(n+1)}. \end{eqnarray*}