L-11

\(11).\;\;\displaystyle \lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{2 n}\right) \)

SOLUTION

\(\displaystyle \Rightarrow \lim _{n \rightarrow \infty}\left(\frac{1}{n}+\frac{1}{n+1}+\cdots+\frac{1}{n+n}\right) \)

\(\displaystyle \Rightarrow \lim _{n \rightarrow \infty} \sum_{k=0}^{n} \frac{1}{n+k} \)

Using Riemann sum Relationship to Definite Integral ,We will get

\(\displaystyle \int_{0}^{1} \frac{d x}{1+x} \)

\(\Rightarrow \displaystyle \left.\ln (1+x)\right|_{0} ^{1} \)

\(\displaystyle \ln 2\)

ANSWER : \(\displaystyle \ln 2\)