Probability+Calculus=Probabulus

Calculus Level pending

A bag contains \(n^2\) black balls and \(1\) white ball.
Now, \(n\) balls are randomly drawn from the bag and replaced in the bag with \(3\) more white balls, then another \(n\) balls are drawn and replaced in the bag with \(5\) more white balls, then \(7\) balls and so on till \(2n-1\) white balls.
If \(P_n\) is the probability that all balls drawn are \(BLACK\), then find \[ \lim_{n \rightarrow \infty}P_n\]

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