A string of pearls

$\begin{aligned} a_2&=1-\frac{J}{a_1} \\ a_3&=1-\frac{J}{a_2}=1-\frac{J}{1-\frac{J}{a_1}} \end{aligned}$

Equivalently, we can write $a_n=1-\frac{J}{a_n}$

So in order to get $a_{2014}$ to be maximum we can take maximum possible value for $a_n$ possible.

so if we extrapolate it to infinity we get,

$a_{\infty}=1-(J/(1-J/(1-J\ldots \text{ infinite terms})))))$

Let the value be $x$ so, we get

$x=1-3/20x$ $\rightarrow 20x^2-20x+3=0$

solving for $x$ , we get $x=(5+\sqrt{10})/10$ and $5=\sqrt(10))/10$

for maximum value we take $(5+\sqrt{10})/10=0.816$