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Question Number 76190 by abdomathmax last updated on 25/Dec/19

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

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 29/Dec/19

1) f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(a r c t a n (1 + x))}^{(k)} \times {(\frac{1}{x + 2})}^{(n - k)}

w e h a v e {(a r c t a n (1 + x))}^{(1)} = \frac{1}{1 + {(1 + x)}^{2}} = \frac{1}{x^{2} + 2 x + 2} \Rightarrow

{(a r c t a n (x + 1))}^{(k)} = {(\frac{1}{x^{2} + 2 x + 2})}^{(k - 1)}

Δ^{^{'}} = 1 - 2 = - 1 \Rightarrow x_{1} = - 1 + i a n d x_{2} = - 1 - i

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

\frac{1}{x^{2} + 2 x + 2} = \frac{1}{(x - \sqrt{2} e^{\frac{i 3 π}{4}}) (x - \sqrt{2} e^{- i \frac{3 π}{4}})} = \frac{1}{\sqrt{2} 2 i s i n (\frac{3 π}{4})} (\frac{1}{x - \sqrt{2} e^{- \frac{i 3 π}{4}}} - \frac{1}{x - \sqrt{2} e^{i \frac{3 π}{4}}})

= \frac{1}{2 i \sqrt{2} \frac{1}{\sqrt{2}}} {\frac{1}{x - \sqrt{2} e^{- \frac{i 3 π}{4}}} - \frac{1}{x - \sqrt{2} e^{\frac{i 3 π}{4}}}} \Rightarrow

\frac{d^{k}}{{d x}^{k}} (a r c t a n (x + 1)) = \frac{1}{2 i} {\frac{{(- 1)}^{k - 1} (k - 1)!}{{(x - \sqrt{2} e^{- \frac{i 3 π}{4}})}^{k}} - \frac{{(- 1)}^{k - 1} (k - 1)!}{{(x - \sqrt{2} e^{\frac{i 3 π}{4}})}^{k}}}

= \frac{{(- 1)}^{k - 1} (k - 1)!}{2 i} {\frac{{(x - \sqrt{2} e^{\frac{i 3 π}{4}})}^{k} - (x - \sqrt{2} e^{- \frac{i 3 π}{4}})}{{(x^{2} + 2 x + 2)}^{k}}}

f^{(n)} (x) = a r c t a n (x + 1) \frac{{(- 1)}^{n} n!}{{(x + 2)}^{n + 1}}

+ \frac{1}{2 i} \sum_{k = 1}^{n} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times \frac{{(x - \sqrt{2} e^{\frac{i 3 π}{4}})}^{k} - {(x - \sqrt{2} e^{- \frac{i 3 π}{4}})}^{k}}{{(x^{2} + 2 x + 2)}^{k}} \times \frac{{(- 1)}^{n - k} (n - k)!}{{(x + 2)}^{n - k + 1}}

= {(- 1)}^{n} n! \frac{a r c t a n (x + 1)}{{(x + 2)}^{n + 1}}

+ {(- 1)}^{n - 1} \sum_{k = 1}^{n} C_{n}^{k} (k - 1)! \frac{I m {{(x - \sqrt{2} e^{i \frac{3 π}{4}})}^{k}}}{{(x^{2} + 2 x + 2)}^{k}} \times \frac{(n - k)!}{{(x + 2)}^{n - k + 1}}

Commented by mathmax by abdo last updated on 29/Dec/19

f^{(n)} (0) = \frac{π}{4} \frac{{(- 1)}^{n} n!}{2^{n + 1}}

+ {(- 1)}^{n - 1} \sum_{k = 1}^{n} C_{n}^{k} \frac{(k - 1)! (n - k)!}{2^{k} \times 2^{n - k + 1}} \times I m {{(- \sqrt{2})}^{k} e^{i \frac{3 k π}{4}}}

= \frac{π}{4} \frac{{(- 1)}^{n} n!}{2^{n + 1}} + {(- 1)}^{n - 1} \sum_{k = 1}^{n} \frac{n!}{k! (n - k)!} \times \frac{(k - 1)! (n - k!}{2^{n + 1}} \times {(- \sqrt{2})}^{k} s i n (\frac{3 k π}{4})

= \frac{π}{4} \times \frac{{(- 1)}^{n} n!}{2^{n + 1}} + \frac{{(- 1)}^{n - 1}}{2^{n + 1}} \sum_{k = 1}^{n} \frac{n!}{k} {(- \sqrt{2})}^{k} s i n (\frac{3 k π}{4})