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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) 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) \dots (α - n + 1) x^{α - n}

s o {x^{\frac{1}{2}}}^{(n)} = \frac{1}{2} (\frac{1}{2} - 1) (\frac{1}{2} - 2) \dots . (\frac{1}{2} - n + 1) x^{\frac{1}{2} - n}

= \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) \dots . (\frac{1 - 2 n + 2)}{2}) x^{\frac{1}{2} - n}

= \frac{1}{2} (- \frac{1}{2}) (- \frac{3}{2}) \dots . (- \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}) \dots . . (- \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}) \dots (- \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}) \dots (- \frac{2 n - 3}{2}) + {(- 1)}^{n} .

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