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Question Number 53957 by maxmathsup by imad last updated on 05/Feb/19

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 by maxmathsup 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} .