letf(x) = ((2x+1)/((x−2)(x^2 +x+1))) 1) calculate f^((n)) (x) 2) find f^((n)) (0) 3) developp f at integr serie. (


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Question Number 38521 by math khazana by abdo last updated on 26/Jun/18

$l e t f (x) = \frac{2 x + 1}{(x - 2) (x^{2} + x + 1)}$ $1) c a l c u l a t e f^{(n)} (x)$ $2) f i n d f^{(n)} (0)$ $3) d e v e l o p p f a t i n t e g r s e r i e .$ $($
Commented by abdo mathsup 649 cc last updated on 28/Jun/18
$1) let drcompose f inside C f(x)= (a/(x−2)) +(b/(x−j)) +(c/(x−j^− )) withj=e^((i2π)/3) f(x)=((2x+1)/((x−2)(x−j)(x−j^− ))) a =lim_(x→2) (x−2)f(x)= (5/7) b=lim_(x→j) (x−j)f(x)= ((2j+1)/((j−2)2i((√3)/2))) =((2j+1)/(i(√3)(j−2))) c=lim_(x→j^− ) (x−j^− )f(x)= ((2j^− +1)/((j^− −2)(−2i((√3)/2)))) =−((2j^− +1)/(i(√3)(j^− −2))) ⇒f(x)= (5/(7(x−2))) +((2j+1)/(i(√3)(j−2)(x−j))) −((2j^(− ) +1)/(i(√3)(j^− −2)(x−j^− ))) ⇒ f^((n)) (x) = (5/7) (((−1)^n n!)/((x−2)^(n+1) )) +((2j+1)/(i(√3)(j−2))) (((−1)^n n!)/((x−j)^(n+1) )) −((2j^− +1)/(i(√3)(j^− −2))) (((−1)^n n!)/((x−j^− )^(n+1) )) but 2)f^((n)) (0) = ((5(−1)^n n!)/(7(−2)^(n+1) )) +((2j+1)/(i(√3)(j−2))) (((−1)^n n!)/((−j)^(n+1) )) −((2j^− +1)/(i(√3)(j^− −2)))(((−1)^n n!)/((−j^− )^(n+1) )) =−(5/7) ((n!)/2^(n+1) ) −((2j+1)/(i(√3)(j−2))) ((n!)/j^(n+1) ) +((2j^− +1)/(i(√3)(j^− −2))) ((n!)/((j^− )^(n+1) )) 3) f(x)=Σ_(n=0) ^∞ ((f^((n)) (0))/(n!)) x^n =Σ_(n=0) ^∞ { −(5/(7 .2^(n+1) )) −((2j+1)/(i(√3)(j−2)j^(n+1) )) +((2j^− +1)/(i(√3)(j^− −2)(j^− )^(n+1) ))}x^n$
$1) l e t d r c o m p o s e f i n s i d e C$ $f (x) = \frac{a}{x - 2} + \frac{b}{x - j} + \frac{c}{x - \bar{j}} w i t h j = e^{\frac{i 2 π}{3}}$ $f (x) = \frac{2 x + 1}{(x - 2) (x - j) (x - \bar{j})}$ $a = {l i m}_{x \to 2} (x - 2) f (x) = \frac{5}{7}$ $b = {l i m}_{x \to j} (x - j) f (x) = \frac{2 j + 1}{(j - 2) 2 i \frac{\sqrt{3}}{2}} = \frac{2 j + 1}{i \sqrt{3} (j - 2)}$ $c = {l i m}_{x \to \bar{j}} (x - \bar{j}) f (x) = \frac{2 \bar{j} + 1}{(\bar{j} - 2) (- 2 i \frac{\sqrt{3}}{2})}$ $= - \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2)} \Rightarrow f (x) = \frac{5}{7 (x - 2)} + \frac{2 j + 1}{i \sqrt{3} (j - 2) (x - j)}$ $- \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2) (x - \bar{j})} \Rightarrow$ $f^{(n)} (x) = \frac{5}{7} \frac{{(- 1)}^{n} n!}{{(x - 2)}^{n + 1}} + \frac{2 j + 1}{i \sqrt{3} (j - 2)} \frac{{(- 1)}^{n} n!}{{(x - j)}^{n + 1}}$ $- \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2)} \frac{{(- 1)}^{n} n!}{{(x - \bar{j})}^{n + 1}} b u t$ $2) f^{(n)} (0) = \frac{5 {(- 1)}^{n} n!}{7 {(- 2)}^{n + 1}} + \frac{2 j + 1}{i \sqrt{3} (j - 2)} \frac{{(- 1)}^{n} n!}{{(- j)}^{n + 1}}$ $- \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2)} \frac{{(- 1)}^{n} n!}{{(- \bar{j})}^{n + 1}}$ $= - \frac{5}{7} \frac{n!}{2^{n + 1}} - \frac{2 j + 1}{i \sqrt{3} (j - 2)} \frac{n!}{j^{n + 1}} + \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2)} \frac{n!}{{(\bar{j})}^{n + 1}}$ $3) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n}$ $= \sum_{n = 0}^{\infty} {- \frac{5}{7 . 2^{n + 1}} - \frac{2 j + 1}{i \sqrt{3} (j - 2) j^{n + 1}} + \frac{2 \bar{j} + 1}{i \sqrt{3} (\bar{j} - 2) {(\bar{j})}^{n + 1}}} x^{n}$
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