User:Arichnad/Tmp

Arichnad (talk · contribs · count · logs · block log · lu · rfa · rfb · arb · rfc · lta · socks)

Arichnad (talk · contribs · deleted · count · AfD · logs · block log · lu · rfar · spi)

Tania Head

Stub moved (copy+paste) to Tania Head.

Math Sandbox

${\begin{aligned}{\pi \over 2}&=\sum _{k=0}^{\infty }{k! \over (2k+1)!!}\\&=1+{1 \over 3}+{1\cdot 2 \over 3\cdot 5}+{1\cdot 2\cdot 3 \over 3\cdot 5\cdot 7}+{1\cdot 2\cdot 3\cdot 4 \over 3\cdot 5\cdot 7\cdot 9}+\cdots \\&=1+{1 \over 3}\left(1+{2 \over 5}\left(1+{3 \over 7}\left(1+{4 \over 9}(1+\cdots )\right)\right)\right)\\&=(1;1,1,1,1,\cdots )_{({1 \over 3},{2 \over 5},{3 \over 7},{4 \over 9},\cdots )}\end{aligned}}$

${\begin{aligned}\pi &=(2;2,2,2,2,\cdots )_{({1 \over 3},{2 \over 5},{3 \over 7},{4 \over 9},\cdots )}\end{aligned}}$

$\sum _{k=1}^{\infty }r^{k}={\frac {r}{1-r}}$

$0.999...=9\times \sum _{k=1}^{\infty }.1^{k}=9\times {\frac {.1}{1-.1}}=1$

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