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Question Number 137112 by Algoritm last updated on 29/Mar/21

Answered by Ñï= last updated on 03/Apr/21

$${\tan }^{-1}\left(\frac{1}{3}\right)+{\tan }^{-1}\left(\frac{2}{9}\right)+{\tan }^{-1}\left(\frac{4}{23}\right)+...$$$$=arg(3+i)+arg(9+2i)+arg(23+4i)+...$$$$=arg[(3+i)(9+2i)(23+4i)\cdot ...]$$$$=arg[(25+15i)(23+4i)\cdot ...]$$$$=arg[(515+445i)\cdot ...]$$$$=arg(x+iy)$$$$={\tan }^{-1}\frac{y}{x}$$

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