Pseudo arithmetic or pseudo geometric?

Calculus Level 2

Which is larger?

A=12+12×2+12×2×2+12×2×2×2+0B=02+12×2+22×2×2+32×2×2×2+\begin{aligned} A &= \dfrac1{2} + \dfrac1{2\times2} + \dfrac1{2\times2\times2} + \dfrac1{2\times2\times2\times2} + \cdots \\ \phantom0\\ B &= \dfrac0{2} + \dfrac1{2\times2} + \dfrac2{2\times2\times2} + \dfrac3{2\times2\times2\times2} + \cdots \end{aligned}

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