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Question Number 106133 by mathmax by abdo last updated on 02/Aug/20

let f (x) = e^{- 2 x} \ln (3 - x^{2})

1) calculate f^{(n)} (x) and f^{(n)} (0)

2) developp f at integr serie

Answered by mathmax by abdo last updated on 03/Aug/20

1) we have f (x) = e^{- 2 x} \ln (3 - x^{2}) \Rightarrow

f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {\ln (3 - x^{2})}^{(k)} {(e^{- 2 x})}^{(n - k)}

= \ln (3 - x^{2}) {(- 2)}^{n} e^{- 2 x} + \sum_{k = 1}^{n} C_{n}^{k} {\ln (3 - x^{2})}^{(k)} {(- 2)}^{n - k} e^{- 2 x}

we have {\ln (3 - x^{2})}^{(1)} = \frac{- 2 x}{3 - x^{2}} = \frac{2 x}{(x - \sqrt{3}) (x + \sqrt{3})}

= \frac{1}{x - \sqrt{3}} + \frac{1}{x + \sqrt{3}} \Rightarrow {\ln (3 - x^{2})}^{(k)} = {(\frac{1}{x - \sqrt{3}} + \frac{1}{x + \sqrt{3}})}^{(k - 1)}

= \frac{{(- 1)}^{k - 1} (\boldsymbol k - 1)!}{{(x - \sqrt{3})}^{k}} + \frac{{(- 1)}^{k - 1} (k - 1)!}{{(x + \sqrt{3})}^{k}}

= {(- 1)}^{k - 1} (k - 1)! {\frac{{(x + \sqrt{3})}^{k} + {(x - \sqrt{3})}^{k}}{{(x^{2} - 3)}^{k}}} \Rightarrow

f^{(n)} (x) = {(- 2)}^{n} e^{- 2 x} \ln (3 - x^{2})

+ \sum_{k = 1}^{n} {(- 2)}^{n - k} e^{- 2 x} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times \frac{{(x + \sqrt{3})}^{k} + {(x - \sqrt{3})}^{k}}{{(x^{2} - 3)}^{k}}

Commented by mathmax by abdo last updated on 03/Aug/20

f^{(n)} (0) = \ln (3) + \sum_{k = 1}^{n} {(- 2)}^{n - k} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times \frac{{(\sqrt{3})}^{k} + {(- \sqrt{3})}^{k}}{{(- 3)}^{k}}

= \ln (3) - \sum_{k = 1}^{n} C_{n}^{k} {(- 1)}^{n - k} 2^{n - k} (k - 1)! \times \frac{{(\sqrt{3})}^{k} + {(- \sqrt{3})}^{k}}{3^{k}}

f^{(n)} (0) = \ln (3) + {(- 2)}^{n} \sum_{k = 1}^{n} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times \frac{{(\sqrt{3})}^{k} + {(- \sqrt{3})}^{k}}{3^{k}}

Commented by mathmax by abdo last updated on 03/Aug/20

2) f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (o)}{n!} x^{n} = \ln (3)

+ \sum_{n = 1}^{\infty} \frac{1}{n!} {\ln (3) + {(- 2)}^{n} \sum_{k = 1}^{n} C_{n}^{k} {(- 1)}^{k - 1} (k - 1)! \times \frac{{(\sqrt{3})}^{k} + {(- \sqrt{3})}^{k}}{3^{k}}} x^{n}

= \ln (3) e^{x} + \sum_{n = 1}^{\infty} {{(- 2)}^{n} \sum_{p = 1}^{[\frac{n}{2}]} C_{n}^{2 p} (- 1) (2 p - 1)! \times \frac{2 . 3^{p}}{3^{2 p}}} x^{n}

f (x) = \ln (3) e^{x} + \sum_{n = 1}^{\infty} {(- 2)}^{n + 1} {\sum_{p = 1}^{[\frac{n}{2}]} C_{n}^{2 p} \frac{(2 p - 1)}{3^{p}}} x^{n}

Commented by abdomathmax last updated on 03/Aug/20

f (x) = \ln (3) \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}

+ \sum_{n = 1}^{\infty} {(- 2)}^{n + 1} {\sum_{p = 1}^{[\frac{n}{2}]} C_{n}^{2 p} \frac{(2 p - 1)!}{3^{p}}} x^{n}