let-f-x-e-x-ln-1-x-2-1-calculate-f-n-0-2-developp-f-at-integr-serie-

Question Number 66795 by mathmax by abdo last updated on 19/Aug/19

l e t f (x) = e^{- x} l n (1 + x^{2})

1) c a l c u l a t e 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 26/Aug/19

1) w e h a v e f (x) = e^{- x} l n (1 + x^{2}) \Rightarrow

f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(l n (1 + x^{2}))}^{(k)} {(e^{- x})}^{(n - k)}

= {(- 1)}^{n} e^{- x} l n (1 + x^{2}) + \sum_{k = 1}^{n} C_{n}^{k} {l n (1 + x^{2})}^{(k)} {(e^{- x})}^{(n - k)}

{(l n (1 + x^{2}))}^{(1)} = \frac{2 x}{1 + x^{2}} \Rightarrow {l n (1 + x^{2})}^{(k)} = {(\frac{2 x}{1 + x^{2}})}^{(k - 1)}

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

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

= {(- 1)}^{k - 1} (k - 1)! \times \frac{2 R e ({(x + i)}^{k})}{{(x^{2} + 1)}^{k}} \Rightarrow

f^{(n)} (x) = {(- 1)}^{n} e^{- x} l n (1 + x^{2})

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

\Rightarrow f^{(n)} (0) = \sum_{k = 1}^{n} C_{n}^{k} 2 {(- 1)}^{k - 1} (k - 1)! c o s (\frac{k π}{2}) {(- 1)}^{n - k}

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

Commented by mathmax by abdo last updated on 26/Aug/19

f^{(n)} (0) = 2 {(- 1)}^{n - 1} \sum_{k = 1}^{n} (k - 1)! C_{n}^{k} c o s (\frac{k π}{2})

Commented by mathmax by abdo last updated on 26/Aug/19

2) w e h a v e f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = f (o) + \sum_{n = 1}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}

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

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

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