Super awesome simplification.

The product becomes $\displaystyle \prod\limits_{n=1}^{2014} \left(1 + \frac{2n+1}{n^{2}}\right) = \prod\limits_{n=1}^{2014} \frac{(n+1)^{2}}{n^{2}} = 2015^{2} = \boxed{4060225}$ ~~~

Try to observe the pattern---

(4).(9/4).(16/9).......(4056196/4052169).(4060225/4056196)

Here,you can observe that---numerator of the terms are same as the denominator of the next term to them,except the last term.So,

(1).(1/1).(1/1)....(1/1)(4060225/1)=4060225

Just observe the pattern. Everything else cancels out except the last term