Binary Operations

Calculus Level 5

Define a binary operation \( a*b = a+b+ab\) , with \( a,b \) are real quantities, where \(*\) denotes the symbol of binary operator.

Now this binary operation is associative, commutative and closed under the set of reals.

Denote \(S = \displaystyle \lim_{n \to \infty} \left ( 1 * \frac {1}{4} * \frac {1}{9} * \ldots * \frac {1}{n^2} \right ) \)

Find \( \lfloor 100000S \rfloor \).

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