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 and Answers Forum

All Questions Topic List
Relation and Functions Questions
Previous in All Question Next in All Question
Previous in Relation and Functions Next in Relation and Functions
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
$1) we have f(x)=Im((1/(1+x))e^(−x+3ix) ) = Im((1/(1+x)) e^((−1+3i)x) )=Im(w(x)) we have w(x)^((n)) ={(1/(1+x)) e^((−1+3i)x) }^((n)) =_(leibniz) Σ_(k=0) ^n C_n ^k ((1/(1+x)))^((k)) (e^((−1+3i)x) )^((n−k)) but (e^(zx) )^((k)) =_(zfixed) z^k e^(zx) ⇒{e^((−1+3i)x) }^((n−k)) =(−1+3i)^(n−k) e^((−1+3i)x) also ((1/(1+x)))^((k)) =(((−1)^k k!)/((1+x)^(k+1) )) ⇒w^((n)) (x) = Σ_(k=0) ^n C_n ^k (((−1)^k k!)/((1+x)^(k+1) )) (−1+3i)^(n−k) e^((−1+3i)x) we have ∣−1+3i∣ =(√(1+9))=(√(10)) ⇒−1+3i =(√(10))(((−1)/(√(10))) +((3i)/(√(10))))=r e^(iθ) ⇒ r=(√(10)) and cosθ =((−1)/(√(10))) , sinθ =(3/(√(10))) ⇒tanθ =−3 ⇒θ =−arctan(3) ⇒ −1+3i =(√(10))e^(−i arctan(3)) =⇒(−1+3i)^(n−k) =10^((n−k)/2) e^(−(n−k)i arctan(3)) and (−1+3i)^(n−k) e^((−1+3i)x) = 10^((n−k)/2) (cos(n−k)arctan3)−isin(n−k)arctan(3)}e^(−x) {cos(3x)+isin(3x)} e^(−x) 10^((n−k)/2) {cos(3x)cos(n−k)arctan3 +isin(3x)cos(n−k)arctan3 −i cos(3x) sin(n−k)arctan3 +sin(3x)sin(n−k)arctan3 ⇒ f^((n)) (x)=Im(w^((n)) )= Σ_(k=0) ^n (−1)^k k! (C_n ^k /((x+1)^(k+1) )) e^(−x) 10^((n−k)/2) { sin(3x)cos{(n−k)arctan(3) −cos(3x) sin{(n−k)arctan(3)} ⇒ f^((n)) (0) =− Σ_(k=0) ^n (−1)^k k! C_n ^k 10^((n−k)/2) sin{(n−k)arctan3}$
$1) w e h a v e f (x) = I m (\frac{1}{1 + x} e^{- x + 3 i x}) = I m (\frac{1}{1 + x} e^{(- 1 + 3 i) x}) = I m (w (x)) w e h a v e$ $w {(x)}^{(n)} = {\frac{1}{1 + x} e^{(- 1 + 3 i) x}}^{(n)} =_{l e i b n i z} \sum_{k = 0}^{n} C_{n}^{k} {(\frac{1}{1 + x})}^{(k)} {(e^{(- 1 + 3 i) x})}^{(n - k)}$ $b u t {(e^{z x})}^{(k)} =_{z f i x e d} z^{k} e^{z x} \Rightarrow {e^{(- 1 + 3 i) x}}^{(n - k)} = {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x} a l s o$ ${(\frac{1}{1 + x})}^{(k)} = \frac{{(- 1)}^{k} k!}{{(1 + x)}^{k + 1}} \Rightarrow w^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} \frac{{(- 1)}^{k} k!}{{(1 + x)}^{k + 1}} {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x}$ $w e h a v e ∣ - 1 + 3 i ∣ = \sqrt{1 + 9} = \sqrt{10} \Rightarrow - 1 + 3 i = \sqrt{10} (\frac{- 1}{\sqrt{10}} + \frac{3 i}{\sqrt{10}}) = r e^{i θ} \Rightarrow$ $r = \sqrt{10} a n d c o s θ = \frac{- 1}{\sqrt{10}}, s i n θ = \frac{3}{\sqrt{10}} \Rightarrow t a n θ = - 3 \Rightarrow θ = - a r c t a n (3) \Rightarrow$ $- 1 + 3 i = \sqrt{10} e^{- i a r c t a n (3)} =\Rightarrow {(- 1 + 3 i)}^{n - k} = 10^{\frac{n - k}{2}} e^{- (n - k) i a r c t a n (3)}$ $a n d {(- 1 + 3 i)}^{n - k} e^{(- 1 + 3 i) x} =$ $10^{\frac{n - k}{2}} (c o s (n - k) a r c t a n 3) - i s i n (n - k) a r c t a n (3)} e^{- x} {c o s (3 x) + i s i n (3 x)}$ $e^{- x} 10^{\frac{n - k}{2}} {c o s (3 x) c o s (n - k) a r c t a n 3 + i s i n (3 x) c o s (n - k) a r c t a n 3$ $- i c o s (3 x) s i n (n - k) a r c t a n 3 + s i n (3 x) s i n (n - k) a r c t a n 3$ $\Rightarrow f^{(n)} (x) = I m (w^{(n)}) =$ $\sum_{k = 0}^{n} {(- 1)}^{k} k! \frac{C_{n}^{k}}{{(x + 1)}^{k + 1}} e^{- x} 10^{\frac{n - k}{2}} {s i n (3 x) c o s {(n - k) a r c t a n (3)$ $- c o s (3 x) s i n {(n - k) a r c t a n (3)} \Rightarrow$ $f^{(n)} (0) = - \sum_{k = 0}^{n} {(- 1)}^{k} k! C_{n}^{k} 10^{\frac{n - k}{2}} s i n {(n - k) a r c t a n 3}$
Commented by maxmathsup by imad last updated on 03/Jun/19
$2) f(x) =Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =Σ_(n=0) ^∞ (1/(n!)){ Σ_(k=0) ^n (−1)^(k+1) k! ((n!)/(k!(n−k)!)) 10^((n−k)/2) sin{(n−k)arctan(3)} =Σ_(n=0) ^∞ (Σ_(k=0) ^n (−1)^(k+1) ((10^((n−k)/2) )/((n−k)!)) sin{(n−k)arctan3})x^n ⇒ f(x) =Σ_(n=0) ^∞ a_n x^n with a_n =Σ_(k=0) ^n (−1)^(k+1) ((10^((n−k)/2) )/((n−k)!)) sin{(n−k)arctan3}$
$2) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} = \sum_{n = 0}^{\infty} \frac{1}{n!} {\sum_{k = 0}^{n} {(- 1)}^{k + 1} k! \frac{n!}{k! (n - k)!} 10^{\frac{n - k}{2}} s i n {(n - k) a r c t a n (3)}$ $= \sum_{n = 0}^{\infty} (\sum_{k = 0}^{n} {(- 1)}^{k + 1} \frac{10^{\frac{n - k}{2}}}{(n - k)!} s i n {(n - k) a r c t a n 3}) x^{n} \Rightarrow$ $f (x) = \sum_{n = 0}^{\infty} a_{n} x^{n} w i t h a_{n} = \sum_{k = 0}^{n} {(- 1)}^{k + 1} \frac{10^{\frac{n - k}{2}}}{(n - k)!} s i n {(n - k) a r c t a n 3}$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum