2 or 3

Algebra Level 3

Which is bigger: \(A\) or \(B\)?

\[\begin{align} A & =\dfrac{2+2^2+2^3+\dots+2^{3n-1}}{\dfrac{1}{2}+\dfrac{1}{2^2}+\dfrac{1}{2^3}+\dots+\dfrac{1}{2^{3n-1}}} \\ \ \\ B & =\dfrac{3+3^2+3^3+\dots+3^{2n-1}}{\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dots+\dfrac{1}{3^{2n-1}}}\end{align}\]

Note: \(n \in \mathbb N\).

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