So Many Geometric Sequences

Algebra Level 3

(1+12+14+18+)×(1+13+19+127+)×(1+14+116+164+)××(1+1n+1n2+)\begin{aligned} &\left( 1 + \dfrac12 + \dfrac14 + \dfrac18 + \cdots\right) \times\left( 1 + \dfrac13 + \dfrac19 + \dfrac1{27} + \cdots \right) \\\\ &\times \left( 1 + \dfrac14 + \dfrac1{16} + \dfrac1{64} + \cdots \right)\times \cdots \times\left( 1 + \dfrac1n + \dfrac1{n^2} + \cdots \right) \end{aligned}

The expression above represents the product of infinite geometric progression sums.

Simplify this expression.

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