"And he came towards a valley, through which ran a river; and the borders of the valley were wooded, and on each side of the river were level meadows. And on one side of the river he saw a flock of white sheep, and on the other a flock of black sheep. And whenever one of the white sheep bleated, one of the black sheep would cross over and become white; and when one of the black sheep bleated, one of the white sheep would cross over and become black."

(Taken from The Mabinogion (1877): Peredur son of Efrawg, p108-109)

Initially each flock consists of $n$ sheep. Each sheep (regardless of colour) is equally likely to be the next sheep to bleat. After a sheep has bleated and a sheep from the other flock has crossed over, Peredur may remove a number of white sheep in order to maximize the expected final number of black sheep. Let $E(n)$ be the expected final number of black sheep if Peredur uses an optimal strategy.

You are given that $E(5) = 6.871346$ rounded to $6$ places behind the decimal point.
Find $E(10\,000)$ and give your answer rounded to $6$ places behind the decimal point.