【 加法展開を繰り返す 】
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加法展開を逐次繰り返すことで、無数の展開式が得られる。下記にその例を示す。
項数が増えるに従って、次の展開式の数は幾何級数的に増大する。

 
      π/4 = arctan(1/1)
                       ｜
                       └→  arctan(1/2)+arctan(1/3)
                                       ｜
                                       ｜
                                       ├→  2arctan(1/2)-arctan(1/7)
                                       ｜              ｜
                                       ｜              ｜
                                       ｜              ├→   arctan(1/2)+arctan(1/3)
                                       ｜              ｜
                                       ｜              ├→   2arctan(1/2)-arctan(1/5)+arctan(1/18)
                                       ｜              ｜
                                       ｜              ├→   2arctan(1/2)-arctan(1/6)+arctan(1/43)
                                       ｜              ｜
                                       ｜              ├→   2arctan(1/2)-arctan(1/8)-arctan(1/57)
                                       ｜              ｜
                                       ｜              ├→   2arctan(1/2)-arctan(1/9)-arctan(1/32)
                                       ｜              ｜
                                       ｜              ├→   2arctan(1/2)-arctan(1/12)-arctan(1/17)
                                       ｜              ｜
                                       ｜              └→   2arctan(1/3)+arctan(1/7)
                                       ｜
                                       ｜
                                       ├→  arctan(1/2)+arctan(1/4)+arctan(1/13)
                                       ｜              ｜
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/3)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/8)-arctan(1/21)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/11)-arctan(1/72)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/12)-arctan(1/157)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/14)+arctan(1/183)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/15)+arctan(1/98)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/18)+arctan(1/47)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/23)+arctan(1/30)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/5)+arctan(1/13)+arctan(1/21)
                                       ｜              ｜
                                       ｜              └→  arctan(1/3)+arctan(1/4)+arctan(1/7)+arctan(1/13)
                                       ｜
                                       ｜
                                       ├→  arctan(1/2)+arctan(1/5)+arctan(1/8)
                                       ｜              ｜
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/3)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/4)+arctan(1/8)-arctan(1/21)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/5)+arctan(1/7)-arctan(1/57)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/5)+arctan(1/9)+arctan(1/73)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/5)+arctan(1/13)+arctan(1/21)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/6)+arctan(1/8)+arctan(1/31)
                                       ｜              ｜
                                       ｜              ├→  arctan(1/2)+arctan(1/7)+arctan(1/8)+arctan(1/18)
                                       ｜              ｜
                                       ｜              └→  arctan(1/3)+arctan(1/5)+arctan(1/7)+arctan(1/8)
                                       ｜
                                       ｜
                                       └→  2arctan(1/3)+arctan(1/7)
                                                         ｜
                                                         ｜
                                                         ├→  arctan(1/2)+arctan(1/3)
                                                         ｜
                                                         ├→  2arctan(1/2)-arctan(1/7)
                                                         ｜
                                                         ├→  2arctan(1/3)+arctan(1/5)-arctan(1/18)
                                                         ｜
                                                         ├→  2arctan(1/3)+arctan(1/6)-arctan(1/43)
                                                         ｜
                                                         ├→  2arctan(1/3)+arctan(1/8)+arctan(1/57)
                                                         ｜
                                                         ├→  2arctan(1/3)+arctan(1/9)+arctan(1/32)
                                                         ｜
                                                         ├→  2arctan(1/3)+arctan(1/12)+arctan(1/17)
                                                         ｜
                                                         ├→  2arctan(1/4)+arctan(1/7)+2arctan(1/13)
                                                         ｜
                                                         └→  2arctan(1/5)+arctan(1/7)+2arctan(1/8)
 
 
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