let-f-x-x-x-2-x-1-1-calculate-f-n-0-2-developp-f-at-integr-serie-

Question Number 36010 by abdo mathsup 649 cc last updated on 27/May/18

l e t f (x) = \frac{x}{x^{2} + x + 1}

1) c a l c u l a t e f^{(n)} (0)

2) d e v e l o p p f a t i n t e g r s e r i e .

Commented by prof Abdo imad last updated on 28/May/18

l e t d e c o m p o s e f (x) i n s i d e C (x) t h e r o o t s o f

p (x) = x^{2} + x + 1 a r e a r e j = e^{i \frac{2 π}{3}} a n d \bar{j} = e^{- i \frac{2 π}{3}} s o

f (x) = \frac{x}{(x - j) (x - \bar{j})} = \frac{a}{x - j} + \frac{b}{x - \bar{j}}

a = {l i m}_{x \to j} (x - j) f (x) = \frac{j}{j - \bar{j}} = \frac{j}{2 \frac{\sqrt{3}}{2}} = \frac{j}{\sqrt{3}}

b = {l i m}_{x \to \bar{j}} (x - \bar{j}) f (x) = \frac{\bar{j}}{\bar{j} - j} = - \frac{\bar{j}}{\sqrt{3}} \Rightarrow

f (x) = \frac{j}{\sqrt{3} (x - j)} - \frac{\bar{j}}{\sqrt{3} (x - \bar{j})} \Rightarrow

f^{(n)} (x) = \frac{j}{\sqrt{3}} {\frac{{(- 1)}^{n} n!}{{(x - j)}^{n + 1}}} - \frac{\bar{j}}{\sqrt{3}} {\frac{{(- 1)}^{n} n!}{{(x - \bar{j})}^{n + 1}}}

f^{(n)} (x) = \frac{{(- 1)}^{n} n!}{\sqrt{3}} {\frac{j}{{(x - j)}^{n + 1}} - \frac{\bar{j}}{{(x - \bar{j})}^{n + 1}}}

= \frac{{(- 1)}^{n} n!}{\sqrt{3}} {\frac{j {(x - \bar{j})}^{n + 1} - \bar{j} {(x - j)}^{n + 1}}{{(x^{2} + x + 1)}^{n + 1}}}

Commented by prof Abdo imad last updated on 28/May/18

s o f^{(n)} (0) = \frac{{(- 1)}^{n} n!}{\sqrt{3}} {\frac{j {(- \bar{j})}^{n + 1} - \bar{j} {(- j)}^{n + 1}}{1}}

= \frac{{(- 1)}^{n} n!}{\sqrt{3}} {{(- 1)}^{n + 1} j {(\bar{j})}^{n + 1} - {(- 1)}^{n + 1} \bar{j} j^{n + 1}}

= \frac{{(- 1)}^{n} n!}{\sqrt{3}} {(- 1)}^{n + 1} {\overset{-^{n}}{j} - j^{n}}

= \frac{1}{\sqrt{3}} {2 I m (j^{n})) = \frac{2}{\sqrt{3}} s i n (\frac{2 n π}{3})

f^{(n)} (0) = \frac{2}{\sqrt{3}} s i n (\frac{2 n π}{3}) .

2) w e h a v e f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} \Rightarrow

f (x) = \sum_{n = 0}^{\infty} \frac{2}{\sqrt{3}} \frac{s i n (\frac{2 n π}{3})}{n!} x^{n} .

f (x) = \frac{2}{\sqrt{3}} \sum_{n = 0}^{\infty} \frac{s i n (\frac{2 n π}{3})}{n!} x^{n} .