Flipping Signs

Algebra Level 2

$\begin{aligned} A &= &1 + \dfrac12 + \dfrac14 + \dfrac18 + \dfrac1{16} +\dfrac1{32} + \dfrac1{64} + \cdots \\ B &=& 1- \dfrac12 + \dfrac14 - \dfrac18 + \dfrac1{16} - \dfrac1{32} + \dfrac1{64} - \cdots \end{aligned}$

What is the relationship between $A$ and $B$ ?