Find $\lim_{n \rightarrow \infty} A_{n}$, where
$A_{n}=\frac{1}{n+1}+\frac{1}{n+2}+ \cdots +\frac{1}{2n}$
with $n \in \mathbb{N}$.
Intuitively, I thought this limit might be zero, nerverthess it isn't. I found in a book (Calculus, Spivak) that this limit can be seen like upper sums and lower sums.
The answer of the problem is that:
$\lim_{n \rightarrow \infty} A_{n}= \int_{0}^{1}\frac{1}{1+x}dx=log(2)$
But I still don't understand how to achieve this result, and how the upper sums and lower sums are involved in the limit of that sequence. Please help me to understand.
