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Question Number 81720 by mathmax by abdo last updated on 14/Feb/20

l e t f (x) = a r c t a n (1 + x^{2})

1) c a l c u l a t e f^{(n)} (x) a n d 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

Commented by mathmax by abdo last updated on 15/Feb/20

1) w e h a v e f^{^{'}} (x) = \frac{2 x}{1 + {(1 + x^{2})}^{2}} = \frac{2 x}{1 + x^{4} + 2 x^{2} + 1} = \frac{2 x}{x^{4} + 2 x^{2} + 2}

x^{4} + 2 x^{2} + 2 = 0 \to t^{2} + 2 t + 2 = 0 (t = x^{2}) \to Δ^{^{'}} = 1 - 2 = - 1 \Rightarrow

x_{1} = - 1 + i = \sqrt{2} e^{i \frac{3 π}{4}} a n d x_{2} = - 1 - i = \sqrt{2} e^{- \frac{i 3 π}{4}} \Rightarrow

f^{^{'}} (x) = \frac{2 x}{(x^{2} - \sqrt{2} e^{\frac{i 3 π}{4}}) (x^{2} - \sqrt{2} e^{- \frac{i 3 π}{4}})} = \frac{2 x}{(x - α e^{\frac{i 3 π}{8}}) (x + α e^{\frac{i 3 π}{8}}) (x - α e^{- \frac{i 3 π}{8}}) (x + α e^{- \frac{i 3 π}{8}} 3}

= \frac{a}{x - α e^{\frac{i 3 π}{8}}} + \frac{b}{x + α e^{\frac{i 3 π}{8}}} + \frac{c}{x - α e^{- i \frac{3 π}{8}}} + \frac{d}{(x + e^{- \frac{i 3 π}{8}})} (α = \sqrt{\sqrt{2}})

a = \frac{2 α e^{\frac{i 3 π}{8}}}{2 α e^{\frac{i 3 π}{8}} \sqrt{2} (2 i) \times \frac{\sqrt{2}}{2}} = \frac{1}{2 i}

b = \frac{- 2 α e^{\frac{i 3 π}{8}}}{- 2 α e^{\frac{i 3 π}{8}} \sqrt{2} (2 i) \times \frac{\sqrt{2}}{2}} = \frac{1}{2 i}

c = \frac{2 α e^{- \frac{i 3 π}{8}}}{2 α e^{- \frac{i 3 π}{8}} \sqrt{2} (- 2 i) \times \frac{\sqrt{2}}{2}} = - \frac{1}{2 i}

d = \frac{- 2 α e^{- \frac{i 3 π}{8}}}{- 2 α e^{- \frac{i 3 π}{8}} \sqrt{2} (- 2 i) \times \frac{\sqrt{2}}{2}} = - \frac{1}{2 i} \Rightarrow

f^{^{'}} (x) = \frac{1}{2 i} {\frac{1}{x - α e^{\frac{i 3 π}{8}}} + \frac{1}{x + α e^{\frac{i 3 π}{8}}} - \frac{1}{x - α e^{- \frac{i 3 π}{8}}} - \frac{1}{x + α e^{- \frac{i 3 π}{8}}}} \Rightarrow

f^{(n)} (x) = \frac{1}{2 i} {\frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - α e^{\frac{i 3 π}{8}})}^{n}} + \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + α e^{\frac{i 3 π}{8}})}^{n}}

- \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x - α e^{- \frac{i 3 π}{8}})}^{n}} - \frac{{(- 1)}^{n - 1} (n - 1)!}{{(x + α e^{- \frac{i 3 π}{8}})}^{n}}}

= \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {\frac{1}{{(x - α e^{\frac{i 3 π}{8}})}^{n}} - \frac{1}{{(x - α e^{- \frac{i 3 π}{8}})}^{n}} + \frac{1}{{(x + α e^{- \frac{i 3 π}{8}})}^{n}}

- \frac{1}{{(x + α e^{- \frac{i 3 π}{8}})}^{n}}}

= \frac{{(- 1)}^{n - 1} (n - 1)!}{2 i} {\frac{- 2 i I m {(x - α e^{- \frac{i 3 π}{8}})}^{n}}{{(x^{2} - 2 α c o s (\frac{3 π}{8}) x + α^{2})}^{n}} + \frac{- 2 i I m {(x + α e^{- \frac{i 3 π}{8}})}^{n}}{{(x^{2} + 2 α c o s (\frac{3 π}{8}) x + α^{2})}^{n}}}

f^{(n)} (x) = {(- 1)}^{n} (n - 1)! {\frac{I m {(x - α e^{- \frac{i 3 π}{8}})}^{n}}{{(x^{2} - 2 α c o s (\frac{3 π}{8}) x + α^{2})}^{n}} + \frac{I m {(x + α e^{- \frac{i 3 π}{8}})}^{n}}{{(x^{2} + 2 α c o s (\frac{3 π}{8}) x + α^{2})}^{n}}}

Commented by mathmax by abdo last updated on 15/Feb/20

f^{(n)} (0) = {(- 1)}^{n} (n - 1)! {\frac{I m {(- α)}^{n} e^{- \frac{i 3 n π}{8}}}{α^{2 n}} + \frac{I m (α^{n} e^{- \frac{i 3 n π}{8}})}{α^{2 n}}

= \frac{{(- 1)}^{n} (n - 1)!}{2^{\frac{n}{2}}} {- {(- α)}^{n} s i n (\frac{3 n π}{8}) - α^{n} s i n (\frac{3 n π}{8})}

= {(- 1)}^{n + 1} (n - 1)! \frac{{2^{\frac{n}{4}} + {(- 1)}^{n} 2^{\frac{n}{4}}} s i n (\frac{3 n π}{8})}{2^{\frac{n}{2}}}

= \frac{{(- 1)}^{n + 1} (n - 1)!}{2^{\frac{n}{4}}} (1 + {(- 1)}^{n}) s i n (\frac{3 n π}{8})