Evaluate $\lim\limits_{n\rightarrow\infty} \left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n}\right)$. Hint: Graph sketching may help.
1. Sketch $y=\frac{x}{x^2-a^2}$ for various values of $a$.
2. Integrate $y=x\ln x$.
3. Evaluate $\lim\limits_{x\to\infty} \big(\ln x-\ln(2x-1)\big).$

What does the graph of $\frac{1}{x}$ look like?

On the above graph, try representing each term of the sum as a (very basic) shape with an area equal to the term’s value.

Which 2 continuous functions can you use as upper and lower bounds for the terms of the sum, having this new representation? Hint: one of them you already used.

Consider the graph below. How do you relate the areas underneath the two functions to the sum?

Squeeze theorem?

Each number $\frac{1}{k}$ is equal to the area of the rectangle extending to the right of the number, having height $\frac{1}{k}$ and width 1. By sketching, notice there are two continuous functions bounding these rectangles, one above, i.e. $\frac{1}{x-1}$, and one below, i.e. $\frac{1}{x}$, whose integrals will also bound the initial sum under the limit.

Concretely,

$$\int_{n+1}^{2n+1} \frac{dx}{x} < \sum_{k=n+1}^{2n}\frac{1}{k} < \int_{n+1}^{2n+1}\frac{dx}{x-1}.$$ At $n\rightarrow\infty$, both integrals become $\ln2,$ and by the squeeze theorem, so must the sum.