let f(x) =arctan(1+2x) 1) calculate f^((n)) (x) then f^((n)) (0) 2) developp f at integr serie . we have f^′ (x)=(2/(1+(1+2x)^2 )) ⇒ f^((n)) (x) =2 {(1/((2x+1)^2 +1))}^((n−1)) with n>0 let W(x)=(1/((2x+1)^2 +1)) ⇒W(x) =(1/((2x+1+i)(2x+1−i))) =(1/(4(x+((1+i)/2))(x+((1−i)/2)))) =(1/(4(x +(1/(√2)) e^((iπ)/4) )( x +(1/(√2)) e^(−((iπ)/4)) ))) but ((1/(x +(1/(√2))e^(−((iπ)/4)) )) −(1/(x+(1/(√2))e^((iπ)/4) ))) =(1/(√2)) (((2isin((π/4))))/((x+(1/(√2))e^(−((iπ)/4)) )(x+(1/(√2))e^((iπ)/4) ))) ⇒ W(x) =(1/(4i)){ (1/(x+(1/(√2))e^(−((iπ)/4)) )) −(1/(x+(1/(√2))e^((iπ)/4) ))} ⇒ W^((n−1)) (x)=(1/(4i)){ (((−1)^(n−1) (n−1)!)/((x+(1/(√2))e^(−((iπ)/4)) )^n )) −(((−1)^(n−1) (n−1)!)/((x+(1/(√2))e^((iπ)/4) )^n ))} ⇒ f^((n)) (x)=(((−1)^(n−1) (n−1)!)/(2i)){ (1/((x+(1/(√2))e^(−((iπ)/4)) )^n )) −(1/((x+(1/(√2))e^((iπ)/4) )^n ))} and f^((n)) (0) =(((−1)^(n−1) (n−1)!)/(2i)){ (1/((√2))^(−n) e^(−i((nπ)/4)) )) −(1/(((√2))^(−n) e^(i((nπ)/4)) ))} =(((−1)^(n−1) (n−1)!)/(2i)){ ((√2))^n e^((inπ)/4) − ((√2))^n e^(−((inπ)/4)) } =(((−1)^(n−1) (n−1)!)/(2i)) ((√2))^n 2i sin(((nπ)/4)) =(−1)^(n−1) (n−1)! ((√2))^n sin(((nπ)/4))


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Question Number 53957 by maxmathsup by imad last updated on 05/Feb/19
$let f(x) =arctan(1+2x) 1) calculate f^((n)) (x) then f^((n)) (0) 2) developp f at integr serie . we have f^′ (x)=(2/(1+(1+2x)^2 )) ⇒ f^((n)) (x) =2 {(1/((2x+1)^2 +1))}^((n−1)) with n>0 let W(x)=(1/((2x+1)^2 +1)) ⇒W(x) =(1/((2x+1+i)(2x+1−i))) =(1/(4(x+((1+i)/2))(x+((1−i)/2)))) =(1/(4(x +(1/(√2)) e^((iπ)/4) )( x +(1/(√2)) e^(−((iπ)/4)) ))) but ((1/(x +(1/(√2))e^(−((iπ)/4)) )) −(1/(x+(1/(√2))e^((iπ)/4) ))) =(1/(√2)) (((2isin((π/4))))/((x+(1/(√2))e^(−((iπ)/4)) )(x+(1/(√2))e^((iπ)/4) ))) ⇒ W(x) =(1/(4i)){ (1/(x+(1/(√2))e^(−((iπ)/4)) )) −(1/(x+(1/(√2))e^((iπ)/4) ))} ⇒ W^((n−1)) (x)=(1/(4i)){ (((−1)^(n−1) (n−1)!)/((x+(1/(√2))e^(−((iπ)/4)) )^n )) −(((−1)^(n−1) (n−1)!)/((x+(1/(√2))e^((iπ)/4) )^n ))} ⇒ f^((n)) (x)=(((−1)^(n−1) (n−1)!)/(2i)){ (1/((x+(1/(√2))e^(−((iπ)/4)) )^n )) −(1/((x+(1/(√2))e^((iπ)/4) )^n ))} and f^((n)) (0) =(((−1)^(n−1) (n−1)!)/(2i)){ (1/((√2))^(−n) e^(−i((nπ)/4)) )) −(1/(((√2))^(−n) e^(i((nπ)/4)) ))} =(((−1)^(n−1) (n−1)!)/(2i)){ ((√2))^n e^((inπ)/4) − ((√2))^n e^(−((inπ)/4)) } =(((−1)^(n−1) (n−1)!)/(2i)) ((√2))^n 2i sin(((nπ)/4)) =(−1)^(n−1) (n−1)! ((√2))^n sin(((nπ)/4))$
$l e t f (x) = a r c t a n (1 + 2 x)$ $1) c a l c u l a t e f^{(n)} (x) t h e n f^{(n)} (0)$ $2) d e v e l o p p f a t i n t e g r s e r i e .$ $w e h a v e f^{^{'}} (x) = \frac{2}{1 + {(1 + 2 x)}^{2}} \Rightarrow f^{(n)} (x) = 2 {\frac{1}{{(2 x + 1)}^{2} + 1}}^{(n - 1)} w i t h n > 0$ $l e t W (x) = \frac{1}{{(2 x + 1)}^{2} + 1} \Rightarrow W (x) = \frac{1}{(2 x + 1 + i) (2 x + 1 - i)}$ $= \frac{1}{4 (x + \frac{1 + i}{2}) (x + \frac{1 - i}{2})} = \frac{1}{4 (x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}}) (x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}})} b u t$ $(\frac{1}{x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}}} - \frac{1}{x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}}}) = \frac{1}{\sqrt{2}} \frac{(2 i s i n (\frac{π}{4}))}{(x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}}) (x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}})} \Rightarrow$ $W (x) = \frac{1}{4 i} {\frac{1}{x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}}} - \frac{1}{x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}}}} \Rightarrow$ $W^{(n - 1)} (x) = \frac{1}{4 i} {\frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}})}^{n}} - \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}})}^{n}}} \Rightarrow$ $f^{(n)} (x) = \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {\frac{1}{{(x + \frac{1}{\sqrt{2}} e^{- \frac{i π}{4}})}^{n}} - \frac{1}{{(x + \frac{1}{\sqrt{2}} e^{\frac{i π}{4}})}^{n}}} a n d$ $f^{(n)} (0) = \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {\frac{1}{{\sqrt{2})}^{- n} e^{- i \frac{n π}{4}}} - \frac{1}{{(\sqrt{2})}^{- n} e^{i \frac{n π}{4}}}}$ $= \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {{(\sqrt{2})}^{n} e^{\frac{i n π}{4}} - {(\sqrt{2})}^{n} e^{- \frac{i n π}{4}}}$ $= \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {(\sqrt{2})}^{n} 2 i s i n (\frac{n π}{4}) = {(- 1)}^{n - 1} (n - 1)! {(\sqrt{2})}^{n} s i n (\frac{n π}{4})$
Commented bymaxmathsup by imad last updated on 05/Feb/19

$2) w e h a v e f (x) = \sum_{n = 0}^{\infty} \frac{x^{n}}{n!} f^{(n)} (0) = \frac{π}{4} + \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} \frac{{(\sqrt{2})}^{n}}{n} s i n (\frac{n π}{4}) x^{n} .$
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