Answer
$\dfrac{596}{105}$
Work Step by Step
We calculate the given sum by expanding it: we calculate each term for $i=1,2,3,4$ and add them:
$$\begin{align*}
\sum_{i=1}^4 \dfrac{2i}{2i-1}&=2+\dfrac{4}{3}+\dfrac{6}{5}+\dfrac{8}{7}\\
&=2\dfrac{386}{105}\\
&=\dfrac{596}{105}.
\end{align*}$$
