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Question Number 81431 by abdomathmax last updated on 13/Feb/20

l e t f (x) = \frac{a r c t a n (2 x)}{1 + 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 abdomathmax last updated on 20/Feb/20

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

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

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

w e h a v e {(a r c t a n (2 x))}^{(1)} = \frac{2}{1 + 4 x^{2}} \Rightarrow

{(a r c t a n (2 x))}^{(k)} = {(\frac{2}{1 + 4 x^{2}})}^{(k - 1)}

= \frac{1}{2} {\frac{1}{x^{2} + \frac{1}{4}}}^{) k - 1)} = \frac{1}{2} {\frac{1}{(x - \frac{i}{2}) (x + \frac{i}{2})}}^{(k - 1)}

= \frac{1}{2 i} {\frac{1}{x - \frac{i}{2}} - \frac{1}{x + \frac{i}{2}}}^{(k - 1)}

= \frac{1}{2 i} {\frac{{(- 1)}^{k - 1}}{{(x - \frac{i}{2})}^{k}} - \frac{{(- 1)}^{k - 1}}{{(x + \frac{i}{2})}^{k}}}

= \frac{{(- 1)}^{k - 1}}{2 i} {\frac{{(x + \frac{i}{2})}^{k} - {(x - \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k}}}

= {(- 1)}^{k - 1} \times \frac{I m {(x + \frac{i}{2})}^{k}}{{(x^{2} + \frac{1}{4})}^{k}} \Rightarrow

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

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

Commented by abdomathmax last updated on 20/Feb/20

f^{(n)} (0) = {(- 1)}^{n - 1} \sum_{k = 1}^{n} C_{n}^{k} \frac{s i n (\frac{k π}{2})}{2^{k} {(\frac{1}{4})}^{k}}

= {(- 1)}^{n - 1} \sum_{k = 1}^{n} C_{n}^{k} 2^{k} s i n (\frac{k π}{2})

2) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}

= f (0) + \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} \times \frac{1}{n!} {\sum_{k = 1}^{n} C_{n}^{k} 2^{k} s i n (\frac{k π}{2})} x^{n}

= \sum_{n = 1}^{\infty} \frac{{(- 1)}^{n - 1}}{n!} \sum_{k = 1}^{n} \frac{n!}{k! (n - k)!} 2^{k} s i n (\frac{k π}{2}) x^{n}

f (x) = \sum_{n = 1}^{\infty} {(- 1)}^{n - 1} (\sum_{k = 1}^{n} \frac{2^{k} s i n (\frac{k π}{2})}{k! (n - k)!}) x^{n}