Qualcosa del genere?
codice:
public class PIApprox {

    // Scegli la precisione
    static double approx = 10E-5;

    public static void main (String[] args) {
        double previous = 4;
        double current = 0;        
        int i = 1;
        while (Math.abs(previous-current) / 2 > approx) {
            current = previous + (4d/(2*i+1))*(i % 2 == 0 ? 1 : -1);            
            i++;
            previous = current;
            current = previous + (4d/(2*i+1))*(i % 2 == 0 ? 1 : -1);
            //System.out.println(previous+"\t\t"+current);
        }
        //System.out.println("\n\n***************************************\n\n");
        System.out.println("Iterazioni: "+i+"\tValore: "+(current+previous)/2);
    }
}