Is this limit related to the harmonic series?

Calculus Level 3

\[ A=\lim_{n\to\infty}{\left(\dfrac{1}{n+1}+\dfrac{1}{n+2}+\cdots+\dfrac{1}{2n} \right)} \]

Find the exact value of \(e^A\) if the limit exists, otherwise enter 0 as your answer.