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$\displaystyle {\begin{alignedat}{8}1&+{\frac {1}{2}}&&+\Big({\frac {1}{3}}&&+{\frac {1}{4}}\Big)&&+\Big({\frac {1}{5}}&&+{\frac {1}{6}}&&+{\frac {1}{7}}&&+{\frac {1}{8}}\Big)&&+\Big({\frac {1}{9}}&&+\cdots \\ {}\geq 1&+{\frac {1}{2}}&&+\Big({\frac {1}{\color {red}{\mathbf {4} }}}&&+{\frac {1}{4}}\Big)&&+\Big({\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{\color {red}{\mathbf {8} }}}&&+{\frac {1}{8}}\Big)&&+\Big({\frac {1}{\color {red}{\mathbf {16} }}}&&+\cdots \\ {}= 1&+{\frac {1}{2}}&& +\ \ {\frac {1}{2}}&& && +\ \ {\frac {1}{2}}&& && && &&+\ \ \ {\frac {1}{2}}&&+\cdots =\infty\end{alignedat}}$ Mercator series $\displaystyle \ln 2 = 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots$ Leibniz Series for $

Displacement $\vec{x(t)}$, Velocity $\vec{v(t)}$, and Acceleration $\vec{a(t)}$: $\lim\limits_{\Delta t\to 0}\frac{\vec{\Delta x}}{\Delta t} = \vec{v} = \int_{t_i}^{t_f}\vec{a}dt$ Uniform Accelerated Motion ($\vec{a}$ has constant magnitude, direction) $\boxed{5-3-2}$ $\displaystyle \vec{\Delta x} = \vec{v_i}\Delta t+\frac{1}{2}\vec{a}(\Delta t)^2 = \vec{v_f}\Delta t-\frac{1}{2}\vec{a}(\Delta t)^2$ $\displaystyle{\vec{v_f^2} = \vec{v_i^2}+2\vec{a}\vec{\Delta{x}}\Leftarrow \begin{cases}\vec{v}_f = \vec{v}_i+\vec{a}\Delta t\\ \vec{\Delta x} = \Big(\frac{\vec{v_i}+\vec{v_f}}{2}\Big) \Delta t\end{cases}}$

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