Find Sequence, get the series

Algebra Level 4

The sequence \(\left\{ { x }_{ n } \right\} \) is defined by \(x_1 = \dfrac{1}{2}\) and \(x_{k+1} = x_k + x_k^2\) for \(k \ge 2\). Then find

\[\left\lfloor \frac { 1 }{ { x }_{ 1 }+1 } +\frac { 1 }{ { x }_{ 2 }+1 } +\cdots +\frac { 1 }{ { x }_{ 100 }+1 } \right\rfloor \]

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