Harmonic Series

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

Mercator series

$\displaystyle \ln 2 = 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-\cdots$

Leibniz Series for $\pi$

$\displaystyle \arctan (1) = \frac {\pi }{4} =\sum _{k=0}^{\infty }{\frac {(-1)^{k}}{2k+1}}=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots $
$\displaystyle \arctan(x)=\int_0^x{\frac {1}{1+x^{2}}}dx=\sum_{n=0}^{\infty}{\frac{(-1)^{n}}{2n+1}}x^{2n+1}=x-\frac{x^3}{3}+\frac{x^5}{5}-\frac{x^7}{7}+\cdots$


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