Is the statement always true?

Let $s_n=\sum_{n+1}^{2n} \frac{1}{x}$

Then $s_1=\frac12$ and for all $n>1$ , $s_{n-1}=\sum_n^{2n-2} \frac{1}{x}$

so that $s_n-s_{n-1}=\frac{1}{2n-1}+\frac{1}{2n}-\frac{1}{n}=\frac{1}{2n-1}-\frac{1}{2n}$

Since $\frac{1}{2n-1}-\frac{1}{2n}>0$ for all $n>1$ , the sequence $s_n$ is increasing; since $s_2=\frac{7}{12}=\frac{14}{24}>\frac{13}{24}$ , the statement is always true .

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Bonus question: what happens to $s_n$ as $n \to \infty$ ?