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I need some help in trying to solve this problem.

We have the sequence defined as $$s_n = \frac{1}{n} + \frac{1}{n+1} + \cdots + \frac{1}{2n}$$ for $n \geq 1$. We have to prove: $$\lim_{n\to \infty} s_n = \ln(2) $$

Now, I tried to rewrite the sequence as $\sum_{k=0}^{n} \frac{1}{n+k}$. And now I am trying to find $a_n$ and $b_n$ such that $a_n \leq s_n \leq b_n $. If the limit of both $a_n$ and $b_n$ tend to $\ln(2)$, then $s_n$ tends to $\ln(2)$ by Squeeze Theorem.

I am having problems to find $a_n$ and $b_n$, and I would like to receive help on this.

Thanks!

Sangchul Lee
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    Where did you get this problem? Is it from a class? If so, I'd recommend you share the particular class (calculus 2, intermediate real analysis, etc.) to ensure the hints/answers you receive use tools and results that are within your scope of knowledge. – Alann Rosas Apr 02 '23 at 02:55

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