### 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 $

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

However, you'll not receive the stake amount, the rationale that} casino paid it for you. Choose a casino which has favorable bonus terms, and ensure it has a good status amongst lovers. – This is a reward to a player who refers a 1xbet pal to the platform who then also indicators up as a member. Scores Casino has separate mobile apps available for iPhone and Android. You also can play with out downloading an app, using your device’s browser.

ReplyDeleteWe can’t give full marks here outcome of|as a outcome of} this online casino solely has 8 fee strategies select from|to choose from}. That mentioned, 1xbet with debit/credit playing cards and Bitcoin, Bitcoin Cash, and credit/debit playing cards, you most likely won’t need the rest. The reside vendor version can also be|can be} super-slick, with European, American, and Auto Roulette options – however keep in mind the min inside bet here is $1. The Auto Roulette version in the Red reside casino also gets way more spins per hour than the common wheel – factors throughout. As a outcome, our roulette video games are frequently examined and certified by unbiased auditors to ensure they pay out what they’re supposed to. Online roulette is the enduring casino desk recreation that everyone conscious of}, whose juicy payouts deliver luggage of enjoyable.

ReplyDelete