let f(x)=(√x)+(1/(x−1)) 1) calculate f^((n)) (2) 2) if f(x) =Σ_(n=0) ^∞ a_n (x−2)^n find the sequence a


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Question Number 42999 by abdo.msup.com last updated on 06/Sep/18

$l e t f (x) = \sqrt{x} + \frac{1}{x - 1}$ $1) c a l c u l a t e f^{(n)} (2)$ $2) i f f (x) = \sum_{n = 0}^{\infty} a_{n} {(x - 2)}^{n} f i n d t h e$ $s e q u e n c e a_{n}$
Commented by maxmathsup by imad last updated on 08/Sep/18
$1) first let find f^((n)) (x) we have f^((n)) (x)=((√x))^n +((1/(x−1)))^n but {(1/(x−1))}^((n)) = (((−1)^n n!)/((x−1)^(n+1) )) let determine ((√x))^n ={x^(1/2) }^((n)) let α fromQ {x^α }^((1)) =α x^(α−1) ⇒{x^α }^((2)) =α(α−1)x^(α−2) ⇒{x^α }^((n)) =α(α−1)...(α−n+1)x^(α−n) so { x^(1/2) }^((n)) =(1/2)((1/2)−1)((1/2)−2)....((1/2) −n+1)x^((1/2)−n) =(1/2)(−(1/2))(−(3/2))....(((1−2n+2))/2))x^((1/2)−n) =(1/2)(−(1/2))(−(3/2))....(−((2n−3)/2)) x^((1/2)−n) ⇒ f^((n)) (x)=(1/2)(−(1/2))(−(3/2))=(−((2n−3)/2))x^((1/2)−n) + (((−1)^n n!)/((x−1)^(n+1) )) .⇒ f^((n)) (2) =(1/2)(−(1/2))(−(3/2)).....(−((2n−3)/2))2^((1/2)−n) +(−1)^n n! = (1/(2^n (√2))) (−(1/2))(−(3/2))...(−((2n−3)/2)) +(−1)^n n! . 2) developpement of f(x) at integr serie seris at v(2) give f(x) =Σ_(n=0) ^∞ ((f^((n)) (2))/(n!)) (x−2)^n ⇒a_n =((f^((n)) (2))/(n!)) ⇒ a_n = (1/((n!)2^n (√2)))(−(1/2))(−(3/2))...(−((2n−3)/2)) +(−1)^n . nt$
$1) f i r s t l e t f i n d f^{(n)} (x) w e h a v e f^{(n)} (x) = {(\sqrt{x})}^{n} + {(\frac{1}{x - 1})}^{n} b u t$ ${\frac{1}{x - 1}}^{(n)} = \frac{{(- 1)}^{n} n!}{{(x - 1)}^{n + 1}} l e t d e t e r m i n e {(\sqrt{x})}^{n} = {x^{\frac{1}{2}}}^{(n)} l e t α f r o m Q$ ${x^{α}}^{(1)} = α x^{α - 1} \Rightarrow {x^{α}}^{(2)} = α (α - 1) x^{α - 2} \Rightarrow {x^{α}}^{(n)} = α (α - 1) . . . (α - n + 1) x^{α - n}$ $s o {x^{\frac{1}{2}}}^{(n)} = \frac{1}{2} (\frac{1}{2} - 1) (\frac{1}{2} - 2) . . . . (\frac{1}{2} - n + 1) x^{\frac{1}{2} - n}$ $= \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) . . . . (\frac{1 - 2 n + 2)}{2}) x^{\frac{1}{2} - n}$ $= \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) . . . . (- \frac{2 n - 3}{2}) x^{\frac{1}{2} - n} \Rightarrow$ $f^{(n)} (x) = \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) = (- \frac{2 n - 3}{2}) x^{\frac{1}{2} - n} + \frac{{(- 1)}^{n} n!}{{(x - 1)}^{n + 1}} . \Rightarrow$ $f^{(n)} (2) = \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) . . . . . (- \frac{2 n - 3}{2}) 2^{\frac{1}{2} - n} + {(- 1)}^{n} n!$ $= \frac{1}{2^{n} \sqrt{2}} (- \frac{1}{2}) (- \frac{3}{2}) . . . (- \frac{2 n - 3}{2}) + {(- 1)}^{n} n! .$ $2) d e v e l o p p e m e n t o f f (x) a t i n t e g r s e r i e s e r i s a t v (2) g i v e$ $f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (2)}{n!} {(x - 2)}^{n} \Rightarrow a_{n} = \frac{f^{(n)} (2)}{n!} \Rightarrow$ $a_{n} = \frac{1}{(n!) 2^{n} \sqrt{2}} (- \frac{1}{2}) (- \frac{3}{2}) . . . (- \frac{2 n - 3}{2}) + {(- 1)}^{n} .$ $n t$
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