let-f-x-1-e-x-n-1-calculate-f-p-x-and-f-p-o-2-calculate-f-n-0-3-developp-f-at-integr-serie-

Question Number 37901 by math khazana by abdo last updated on 19/Jun/18

l e t f (x) = {(1 + e^{- x})}^{n}

1) c a l c u l a t e f^{(p)} (x) a n d f^{(p)} (o)

2) c a l c u l a t e 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 math khazana by abdo last updated on 21/Jun/18

w e h a v e f (x) = \sum_{k = 0}^{n} C_{n}^{k} e^{- k x} \Rightarrow

f^{(p)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(e^{- k x})}^{(p)} b u t

{(e^{- k x})}^{(1)} = - k e^{- k x} a n d {(e^{- k x})}^{(2)} = {(- k^{})}^{2} e^{- k x}

{(e^{- k x})}^{(p)} = {(- k)}^{p} e^{- k x} \Rightarrow

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

f^{(p)} (0) = \sum_{k = 0}^{n} {(- k)}^{p} C_{n}^{k}

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

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

= \sum_{p = 0}^{\infty} \frac{1}{p!} {\sum_{k = 0}^{n} {(- k)}^{p} C_{n}^{k}} x^{p} .