f-x-x-2-e-x-f-100-0-

Question Number 81439 by jagoll last updated on 13/Feb/20

f (x) = x^{2} e^{x}

f^{(100)} (0) = ?

Commented by john santu last updated on 13/Feb/20

\Rightarrow f (x) = x^{2} e^{x} = \underset{n = 0}{\sum^{\infty}} \frac{x^{n + 2}}{n!}

\Rightarrow n + 2 = 100 \Rightarrow f (x) = \underset{n = 0}{\sum^{98}} \frac{x^{100}}{98!}

f^{(100)} (0) = \frac{100!}{98!} = 100 \times 99 = 9900

Commented by abdomathmax last updated on 13/Feb/20

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

= C_{n}^{0} x^{2} e^{x} + 2 x C_{n}^{1} e^{x} + 2 C_{n}^{2} e^{x}

= x^{2} e^{x} + 2 n x e^{x} + 2 \times \frac{n!}{2! (n - 2)!} e^{x}

= (x^{2} + 2 n x + n (n - 1)) e^{x} \Rightarrow

f^{(100)} (0) = 100 \times 99 = 9900