Given a series:
$$\frac{1}n+\frac{1}{n+1}+\cdots+\frac{1}{2n-1}$$
What are these types of questions called and what is the strategy for them?
The next step in the solution manual is:
$$\frac{1}{n} \left(1+\frac{1}{1+\frac{1}{n}}+\cdots+\frac{1}{1+\frac{n-1}{n}}\right)$$
And final answer for this specific question is:
$$\int_1^2 \frac{1}{x} \rightarrow \left[\ln x \vphantom{\frac 11} \right]_1^2 $$
It's a Riemann sum. It is easy for those who have seen the definition of "Riemann sum" to be unaccustomed to recognizing them when they see them. \begin{align} & \frac{1}n+\frac{1}{n+1}+\cdots+\frac{1}{2n-1} \\[10pt] = {} & \frac 1 n \left( 1 + \frac 1 {1 + \frac 1 n} + \frac 1 {1 + \frac 2 n} + \cdots + \frac 1 {1 + \frac{n-1} n } \right) \\[10pt] = {} & \Delta x \Big( f(1) + f(1+\Delta x) + f(1+2\,\Delta x) + \cdots + f(1 + (n-1)\,\Delta x) \Big) \\[10pt] \to {} & \int_1^2 f(x)\,dx \quad\text{as }\Delta x\to 0 \qquad\text{where }f(x) = \frac 1 x. \end{align}
Notice the proper use of the $\text{“}\to\text{''}$. As I used it, it means "approaches". It can also be correctly used when $A\to B$ means $\text{“If }A\text{ then }B\text{''}$, or when $\text{“}f:A\to B\text{ ''}$ means $f$ is a function that maps the set $A$ into the set $B$. Where you used it you should have had $\text{“}=\text{''}$.