let h(x)= arctan(x+(1/x)) 1)calculate h^((n)) (x) and h^((n)) (1) 2)developp f(x)at integr serie at x


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Question Number 62440 by mathsolverby Abdo last updated on 21/Jun/19

$l e t h (x) = a r c t a n (x + \frac{1}{x})$ $1) c a l c u l a t e h^{(n)} (x) a n d h^{(n)} (1)$ $2) d e v e l o p p f (x) a t i n t e g r s e r i e a t x_{0} = 1$
Commented by mathmax by abdo last updated on 23/Jun/19
$1) we have h^′ (x)=((1−(1/x^2 ))/(1+(x+(1/x))^2 )) =((x^2 −1)/(x^2 +(x^2 +1)^2 )) =((x^2 −1)/(x^2 +x^4 +2x^2 +1)) =((x^2 −1)/(x^4 +3x^2 +1)) x^2 =t ⇒h^′ (x) =g(t) =((t−1)/(t^2 +3t +1)) Δ =9−4 =5 ⇒t_1 =((−3+(√5))/2) and t_2 =((−3−(√5))/2) ⇒g(t) =((t−1)/((t−t_1 )(t−t_2 ))) =((x^2 −1)/((x^2 −((−3+(√5))/2))(x^2 +((3+(√5))/2)))) =((x^2 −1)/((x^2 +((3−(√5))/2))(x^2 +((3+(√5))/2)))) =(x^2 −1)(1/(√5)){ (1/(x^2 +((3−(√5))/2))) −(1/(x^2 +((3+(√5))/2)))} =(1/(√5)){ ((x^2 −1)/(x^2 +((3−(√5))/2))) −((x^2 −1)/(x^2 +((3+(√5))/2)))} =(1/(√5)){ 1−((1+((3−(√5))/2))/(x^2 +((3−(√5))/2))) −(1−((1+((3+(√5))/2))/(x^2 +((3+(√5))/2))))} =((5−(√5))/(2(√5))){ (1/(x^2 +((3−(√5))/2)))} +((5+(√5))/(2(√5))){ (1/(x^2 +((3+(√5))/2)))} let α =(√((3−(√5))/2)) and β =(√((3+(√5))/2)) ⇒h^′ (x)=(((√5)−1)/2){ (1/(x^2 +α^2 )) +(1/(x^2 +β^2 ))} =(((√5)−1)/2){ (1/((x+iα)(x−iα))) +(1/((x−iβ)(x+iβ)))} =(((√5)−1)/2){ (1/(2iα)) ((1/(x−iα)) −(1/(x+iα))) +(1/(2iβ))( (1/(x−iβ)) −(1/(x+iβ)))}⇒ h^((n)) (x) =(((√5)−1)/(4iα)){ ((1/(x−iα)))^((n−1)) −((1/(x+iα)))^(n−1) }+(((√5)−1)/(4iβ)){ ((1/(x−iβ)))^((n−1)) −((1/(x+iβ)))^((n−1)) } =(((√5)−1)/(4iα)){ (((−1)^(n−1) (n−1)!)/((x−iα)^n )) −(((−1)^(n−1) (n−1)!)/((x+iα)^n ))} +(((√5)−1)/(4iβ)){ (((−1)^(n−1) (n−1)!)/((x−iβ)^n )) −(((−1)^(n−1) (n−1)!)/((x+iβ)^n ))} =((((√5)−1)(−1)^(n−1) (n−1)!)/(4i)){(1/α)( (1/((x−iα)^n )) −(1/((x+iα)^n ))) +(1/β)((1/((x−iβ)^n )) −(1/((x+iβ)^n )))} =((((√5)−1)(−1)^(n−1) (n−1)!)/(4i)){((2i)/α) ((Im(x+iα)^n )/((x^2 +α^2 )^n )) +((2i)/β) ((Im(x+iβ)^n )/((x^2 +β^2 )^n ))} h^((n)) (x)=(((√5)−1)/2)(−1)^(n−1) (n−1)!{ ((Im(x+iα)^n )/(α(x^2 +α^2 )^n )) +((Im(x+iβ)^n )/((x^2 +β^2 )^n ))} h^((n)) (1) =(((√5)−1)/2)(−1)^(n−1) (n−1)!{ ((Im(1+iα)^n )/(α(1+α^2 )^n )) +((Im(1+iβ)^n )/((1+β^2 )^n ))} . 3)h(x) =Σ_(n=0) ^∞ ((h^((n)) (1))/(n!))(x−1)^n and h^((n)) (1) is known.$
$1) w e h a v e h^{^{'}} (x) = \frac{1 - \frac{1}{x^{2}}}{1 + {(x + \frac{1}{x})}^{2}} = \frac{x^{2} - 1}{x^{2} + {(x^{2} + 1)}^{2}} = \frac{x^{2} - 1}{x^{2} + x^{4} + 2 x^{2} + 1} = \frac{x^{2} - 1}{x^{4} + 3 x^{2} + 1}$ $x^{2} = t \Rightarrow h^{^{'}} (x) = g (t) = \frac{t - 1}{t^{2} + 3 t + 1}$ $Δ = 9 - 4 = 5 \Rightarrow t_{1} = \frac{- 3 + \sqrt{5}}{2} a n d t_{2} = \frac{- 3 - \sqrt{5}}{2} \Rightarrow g (t) = \frac{t - 1}{(t - t_{1}) (t - t_{2})}$ $= \frac{x^{2} - 1}{(x^{2} - \frac{- 3 + \sqrt{5}}{2}) (x^{2} + \frac{3 + \sqrt{5}}{2})} = \frac{x^{2} - 1}{(x^{2} + \frac{3 - \sqrt{5}}{2}) (x^{2} + \frac{3 + \sqrt{5}}{2})}$ $= (x^{2} - 1) \frac{1}{\sqrt{5}} {\frac{1}{x^{2} + \frac{3 - \sqrt{5}}{2}} - \frac{1}{x^{2} + \frac{3 + \sqrt{5}}{2}}} = \frac{1}{\sqrt{5}} {\frac{x^{2} - 1}{x^{2} + \frac{3 - \sqrt{5}}{2}} - \frac{x^{2} - 1}{x^{2} + \frac{3 + \sqrt{5}}{2}}}$ $= \frac{1}{\sqrt{5}} {1 - \frac{1 + \frac{3 - \sqrt{5}}{2}}{x^{2} + \frac{3 - \sqrt{5}}{2}} - (1 - \frac{1 + \frac{3 + \sqrt{5}}{2}}{x^{2} + \frac{3 + \sqrt{5}}{2}})}$ $= \frac{5 - \sqrt{5}}{2 \sqrt{5}} {\frac{1}{x^{2} + \frac{3 - \sqrt{5}}{2}}} + \frac{5 + \sqrt{5}}{2 \sqrt{5}} {\frac{1}{x^{2} + \frac{3 + \sqrt{5}}{2}}} l e t α = \sqrt{\frac{3 - \sqrt{5}}{2}} a n d β = \sqrt{\frac{3 + \sqrt{5}}{2}}$ $\Rightarrow h^{^{'}} (x) = \frac{\sqrt{5} - 1}{2} {\frac{1}{x^{2} + α^{2}} + \frac{1}{x^{2} + β^{2}}}$ $= \frac{\sqrt{5} - 1}{2} {\frac{1}{(x + i α) (x - i α)} + \frac{1}{(x - i β) (x + i β)}}$ $= \frac{\sqrt{5} - 1}{2} {\frac{1}{2 i α} (\frac{1}{x - i α} - \frac{1}{x + i α}) + \frac{1}{2 i β} (\frac{1}{x - i β} - \frac{1}{x + i β})} \Rightarrow$ $h^{(n)} (x) = \frac{\sqrt{5} - 1}{4 i α} {{(\frac{1}{x - i α})}^{(n - 1)} - {(\frac{1}{x + i α})}^{n - 1}} + \frac{\sqrt{5} - 1}{4 i β} {{(\frac{1}{x - i β})}^{(n - 1)} - {(\frac{1}{x + i β})}^{(n - 1)}}$ $= \frac{\sqrt{5} - 1}{4 i α} {\frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - i α)}^{n}} - \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + i α)}^{n}}}$ $+ \frac{\sqrt{5} - 1}{4 i β} {\frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - i β)}^{n}} - \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + i β)}^{n}}}$ $= \frac{(\sqrt{5} - 1) {(- 1)}^{n - 1} (n - 1)!}{4 i} {\frac{1}{α} (\frac{1}{{(x - i α)}^{n}} - \frac{1}{{(x + i α)}^{n}}) + \frac{1}{β} (\frac{1}{{(x - i β)}^{n}} - \frac{1}{{(x + i β)}^{n}})}$ $= \frac{(\sqrt{5} - 1) {(- 1)}^{n - 1} (n - 1)!}{4 i} {\frac{2 i}{α} \frac{I m {(x + i α)}^{n}}{{(x^{2} + α^{2})}^{n}} + \frac{2 i}{β} \frac{I m {(x + i β)}^{n}}{{(x^{2} + β^{2})}^{n}}}$ $h^{(n)} (x) = \frac{\sqrt{5} - 1}{2} {(- 1)}^{n - 1} (n - 1)! {\frac{I m {(x + i α)}^{n}}{α {(x^{2} + α^{2})}^{n}} + \frac{I m {(x + i β)}^{n}}{{(x^{2} + β^{2})}^{n}}}$ $h^{(n)} (1) = \frac{\sqrt{5} - 1}{2} {(- 1)}^{n - 1} (n - 1)! {\frac{I m {(1 + i α)}^{n}}{α {(1 + α^{2})}^{n}} + \frac{I m {(1 + i β)}^{n}}{{(1 + β^{2})}^{n}}} .$ $3) h (x) = \sum_{n = 0}^{\infty} \frac{h^{(n)} (1)}{n!} {(x - 1)}^{n} a n d h^{(n)} (1) i s k n o w n .$
Commented by mathmax by abdo last updated on 23/Jun/19
$error of typo h^((n)) (x) =(((√5)−1)/2)(−1)^(n−1) (n−1)!{ ((Im(x+iα)^n )/(α(x^2 +α^2 )^n )) +((Im(x+iβ)^n )/(β(x^2 +β^2 )^n ))}.$
$e r r o r o f t y p o$ $h^{(n)} (x) = \frac{\sqrt{5} - 1}{2} {(- 1)}^{n - 1} (n - 1)! {\frac{I m {(x + i α)}^{n}}{α {(x^{2} + α^{2})}^{n}} + \frac{I m {(x + i β)}^{n}}{β {(x^{2} + β^{2})}^{n}}} .$
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