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)
Stub moved (copy+paste) to Tania Head.
π 2 = ∑ k = 0 ∞ k ! ( 2 k + 1 ) ! ! = 1 + 1 3 + 1 ⋅ 2 3 ⋅ 5 + 1 ⋅ 2 ⋅ 3 3 ⋅ 5 ⋅ 7 + 1 ⋅ 2 ⋅ 3 ⋅ 4 3 ⋅ 5 ⋅ 7 ⋅ 9 + ⋯ = 1 + 1 3 ( 1 + 2 5 ( 1 + 3 7 ( 1 + 4 9 ( 1 + ⋯ ) ) ) ) = ( 1 ; 1 , 1 , 1 , 1 , ⋯ ) ( 1 3 , 2 5 , 3 7 , 4 9 , ⋯ ) {\displaystyle {\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}}}
π = ( 2 ; 2 , 2 , 2 , 2 , ⋯ ) ( 1 3 , 2 5 , 3 7 , 4 9 , ⋯ ) {\displaystyle {\begin{aligned}\pi &=(2;2,2,2,2,\cdots )_{({1 \over 3},{2 \over 5},{3 \over 7},{4 \over 9},\cdots )}\end{aligned}}}
∑ k = 1 ∞ r k = r 1 − r {\displaystyle \sum _{k=1}^{\infty }r^{k}={\frac {r}{1-r}}}
0.999... = 9 × ∑ k = 1 ∞ .1 k = 9 × .1 1 − .1 = 1 {\displaystyle 0.999...=9\times \sum _{k=1}^{\infty }.1^{k}=9\times {\frac {.1}{1-.1}}=1}
