Answer
$$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right)=3$$
Work Step by Step
Given $$\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right)$$
So, we get
\begin{align}
L&=\lim _{n \rightarrow \infty} \sum_{i=1}^{n}\left(1+\frac{i}{n}\right)\left(\frac{2}{n}\right)\\
&=2 \lim _{n \rightarrow \infty} \frac{1}{n}\left[\sum_{i=1}^{n} 1+\frac{1}{n} \sum_{n=1}^{n} i\right]\\
\end{align}
Since $$\sum_{i=1}^{n} 1=n, \ \ \ \sum_{n=1}^{n} i=\frac{n(n+1)}{2}$$
so, we get
\begin{align}
L&=2 \lim _{n \rightarrow \infty} \frac{1}{n}\left[n+\frac{1}{n}\left(\frac{n(n+1)}{2}\right)\right]\\
&
=2 \lim _{n \rightarrow \infty}\left[1+\frac{n^{2}+n}{2 n^{2}}\right]\\
&
=2 \lim _{n \rightarrow \infty}\left[1+\frac{1}{2}+\frac{1}{2n}\right]\\
&=2\left(1+\frac{1}{2}\right), \ \ \ \ \ (as \lim _{n \rightarrow \infty}\frac{1}{n}=0) \\
&=3
\end{align}
