let-f-x-e-x-x-2-4-1-calculate-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-

Question Number 46424 by maxmathsup by imad last updated on 25/Oct/18

l e t f (x) = \frac{e^{- x}}{x^{2} + 4}

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 maxmathsup by imad last updated on 27/Oct/18

1) w e h a v e f (x) = \frac{e^{- x}}{(x - 2 i) (x + 2 i)} = \frac{1}{4 i} e^{- x} {\frac{1}{x - 2 i} - \frac{1}{x + 2 i}}

= \frac{1}{4 i} {\frac{e^{- x}}{x - 2 i} - \frac{e^{- x}}{x + 2 i}} \Rightarrow f^{(n)} (x) = \frac{1}{4 i} {{(\frac{e^{- x}}{x - 2 i})}^{(n)} - {(\frac{e^{- x}}{x + 2 i})}^{(n)}}

l e i b n i z f o r m u l a e g i v e

{(\frac{e^{- x}}{x - 2 i})}^{(n)} = \sum_{k = 0}^{n} C_{n}^{k} {(\frac{1}{x - 2 i})}^{(k)} {(e^{- x})}^{(n - k)}

= \sum_{k = 0}^{n} C_{n}^{k} \frac{{(- 1)}^{k} k!}{{(x - 2 i)}^{k + 1}} {(- 1)}^{n - k} e^{- x} a l s o w e h a v e

{(\frac{e^{- x}}{x + 2 i})}^{(n)} = \sum_{k = 0}^{n} C_{n}^{k} \frac{{(- 1)}^{k} k!}{{(x + 2 i)}^{k + 1}} {(- 1)}^{n - k} e^{- x} \Rightarrow

f^{(n)} (x) = \frac{1}{4 i} \sum_{k = 0}^{n} {(- 1)}^{n} C_{n}^{k} k! {\frac{1}{{(x - 2 i)}^{k + 1}} - \frac{1}{{(x + 2 i)}^{k + 1}}} e^{- x}

= \frac{1}{4 i} \sum_{k = 0}^{n} {(- 1)}^{n} \frac{n!}{(n - k)!} \frac{{(x + 2 i)}^{k + 1} - {(x - 2 i)}^{k + 1}}{{(x^{2} + 4)}^{k + 1}} e^{- x}

Commented by maxmathsup by imad last updated on 27/Oct/18

x = 0 \Rightarrow f^{(n)} (0) = \frac{1}{4 i} \sum_{k = 0}^{n} \frac{{(- 1)}^{n} n!}{(n - k)!} \frac{{(2 i)}^{k + 1} - {(- 2 i)}^{k + 1}}{4^{k + 1}}

= \frac{1}{4 i} \sum_{k = 0}^{n} \frac{{(- 1)}^{n} n!}{(n - k)!} \frac{2 i I m {{(2 i)}^{k + 1}}}{4^{k + 1}} = \frac{1}{2} \sum_{k = 0}^{n} \frac{{(- 1)}^{n} n!}{(n - k)!} \frac{2^{k + 1} s i n (\frac{(k + 1) π}{2})}{2^{2 k + 2}}

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

Commented by maxmathsup by imad last updated on 27/Oct/18

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

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