let-f-x-e-x-1-x-sin-3x-1-dtermine-f-n-x-and-f-n-0-2-developp-f-at-integr-serie-

Question Number 61329 by maxmathsup by imad last updated on 01/Jun/19

l e t f (x) = \frac{e^{- x}}{1 + x} s i n (3 x)

1) d 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 .

Commented by maxmathsup by imad last updated on 03/Jun/19

1) w e h a v e f (x) = I m (\frac{1}{1 + x} e^{- x + 3 i x}) = I m (\frac{1}{1 + x} e^{(- 1 + 3 i) x}) = I m (w (x)) w e h a v e

w {(x)}^{(n)} = {\frac{1}{1 + x} e^{(- 1 + 3 i) x}}^{(n)} =_{l e i b n i z} \sum_{k = 0}^{n} C_{n}^{k} {(\frac{1}{1 + x})}^{(k)} {(e^{(- 1 + 3 i) x})}^{(n - k)}

b u t {(e^{z x})}^{(k)} =_{z f i x e d} z^{k} e^{z x} \Rightarrow {e^{(- 1 + 3 i) x}}^{(n - k)} = {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x} a l s o

{(\frac{1}{1 + x})}^{(k)} = \frac{{(- 1)}^{k} k!}{{(1 + x)}^{k + 1}} \Rightarrow w^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} \frac{{(- 1)}^{k} k!}{{(1 + x)}^{k + 1}} {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x}

w e h a v e ∣ - 1 + 3 i ∣ = \sqrt{1 + 9} = \sqrt{10} \Rightarrow - 1 + 3 i = \sqrt{10} (\frac{- 1}{\sqrt{10}} + \frac{3 i}{\sqrt{10}}) = r e^{i θ} \Rightarrow

r = \sqrt{10} a n d c o s θ = \frac{- 1}{\sqrt{10}}, s i n θ = \frac{3}{\sqrt{10}} \Rightarrow t a n θ = - 3 \Rightarrow θ = - a r c t a n (3) \Rightarrow

- 1 + 3 i = \sqrt{10} e^{- i a r c t a n (3)} =\Rightarrow {(- 1 + 3 i)}^{n - k} = 10^{\frac{n - k}{2}} e^{- (n - k) i a r c t a n (3)}

a n d {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x} =

10^{\frac{n - k}{2}} (c o s (n - k) a r c t a n 3) - i s i n (n - k) a r c t a n (3)} e^{- x} {c o s (3 x) + i s i n (3 x)}

e^{- x} 10^{\frac{n - k}{2}} {c o s (3 x) c o s (n - k) a r c t a n 3 + i s i n (3 x) c o s (n - k) a r c t a n 3

- i c o s (3 x) s i n (n - k) a r c t a n 3 + s i n (3 x) s i n (n - k) a r c t a n 3

\Rightarrow f^{(n)} (x) = I m (w^{(n)}) =

\sum_{k = 0}^{n} {(- 1)}^{k} k! \frac{C_{n}^{k}}{{(x + 1)}^{k + 1}} e^{- x} 10^{\frac{n - k}{2}} {s i n (3 x) c o s {(n - k) a r c t a n (3)

- c o s (3 x) s i n {(n - k) a r c t a n (3)} \Rightarrow

f^{(n)} (0) = - \sum_{k = 0}^{n} {(- 1)}^{k} k! C_{n}^{k} 10^{\frac{n - k}{2}} s i n {(n - k) a r c t a n 3}

Commented by maxmathsup by imad last updated on 03/Jun/19

2) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{1}{n!} {\sum_{k = 0}^{n} {(- 1)}^{k + 1} k! \frac{n!}{k! (n - k)!} 10^{\frac{n - k}{2}} s i n {(n - k) a r c t a n (3)}

= \sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} {(- 1)}^{k + 1} \frac{10^{\frac{n - k}{2}}}{(n - k)!} s i n {(n - k) a r c t a n 3}) x^{n} \Rightarrow

f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} w i t h a_{n} = \sum_{k = 0}^{n} {(- 1)}^{k + 1} \frac{10^{\frac{n - k}{2}}}{(n - k)!} s i n {(n - k) a r c t a n 3}