let-f-x-e-nx-ln-1-x-2-1-determine-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-n-integr-natural-

Question Number 74502 by mathmax by abdo last updated on 25/Nov/19

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

1) d e t e r m i n 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 (n i n t e g r n a t u r a l)

Commented by mathmax by abdo last updated on 25/Nov/19

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

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

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

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

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

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

Commented by mathmax by abdo last updated on 25/Nov/19

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

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

a n d

f^{(n)} (0) = \sum_{k = 1}^{n} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times (i^{k} + {(- i)}^{k}) {(- n)}^{n - k} e^{- n x}

i^{k} + {(- i)}^{k} = 2 R e (i^{k}) = 2 c o s (\frac{k π}{2}) \Rightarrow

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

Commented by mathmax by abdo last updated on 25/Nov/19

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

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

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

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

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

Commented by mathmax by abdo last updated on 25/Nov/19

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

Answered by mind is power last updated on 25/Nov/19

{l n}^{(k)} (1 + x^{2}) = l n (1 + x^{2}), k = 0

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

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

x + i = {\begin{cases} \sqrt{x^{2} + 1} . e^{i a r c t a n (\frac{1}{x})}, x > 0 \\ \sqrt{x^{2} + 1} e^{i (\frac{π}{2} + a r c t a n (\frac{1}{x}))}, x < 0 \end{cases}

x ⩾ 0

\frac{d}{{d x}^{k}} {l n (1 + x^{2})} = \frac{{(- 1)}^{k - 1} (k - 1)! {(x^{2} + 1)}^{\frac{k - 1}{2}} 2 R e {e^{i (k - 1) a r c t a n (\frac{1}{x})}}}{{(x^{2} + 1)}^{k}}, k ⩾ 1

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

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

= \underset{k = 1}{\sum^{n}} C_{n}^{k} {(- n)}^{(n - k)} . e^{- n x} . \frac{{(- 1)}^{k - 1} (k - 1)! . c o s ((k - 1) (a r c t a n (\frac{1}{x}))}{{(x^{2} + 1)}^{\frac{k + 1}{2}}} + {(- n)}^{n} e^{- n x} l n (1 + x^{2})

f^{(n)} (0)

= \underset{k = 1}{\sum^{n}} C_{n}^{k} {(- n)}^{n - k} . {(- 1)}^{k - 1} . (k - 1)! . c o s ((k - 1) \frac{π}{2})

2)

f (x) = \underset{j = 0}{\sum^{+ \infty}} \frac{f^{j} (0) x^{j}}{j!}

f (x) = \sum_{j ⩾ 1} \frac{f^{j} (0) x^{j}}{j!} = \sum_{j ⩾ 1} . (\underset{k = 1}{\sum^{j}} C_{j}^{k} . {(- n)}^{j - k} {(- 1)}^{k - 1} c o s (\frac{k - 1}{2} π)) x^{j} . \frac{1}{j!}