find-residuo-x-21-x-2-x-1-x-1-3-

Question Number 216489 by manxsol last updated on 09/Feb/25

f i n d r e s i d u o

\frac{x^{21} + x^{2} + x + 1}{{(x - 1)}^{3}}

Commented by issac last updated on 09/Feb/25

Q 216493

Answered by MrGaster last updated on 09/Feb/25

Let f (x) = \frac{x^{21} + x^{2} + 1}{{(x - 1)}^{3}}

Residue at x = 1

Let z = x x - 1 \Rightarrow x = z + 1

f (z + 1) = \frac{{(z + 1)}^{21} + {(z + 1)}^{2} + (z + 1) + 1}{z^{3}}

{(z + 1)}^{21} = \underset{k = 0}{\sum^{21}} (\begin{matrix} 21 \\ k \end{matrix}) z^{k}

{(z + 1)}^{2} = z^{2} + 2 z + 1

(z + 1) = z + 1

f (z + 1) = \frac{\underset{k = 0}{\sum^{21}} (\begin{matrix} 21 \\ k \end{matrix}) z^{k} + z^{2} + 2 z + 1 + z + 1 + 1}{z^{3}}

f (z + 1) = \frac{\underset{k = 0}{\sum^{21}} (\begin{matrix} 21 \\ k \end{matrix}) z^{k} + z^{2} + 3 z + 3}{z^{3}}

z^{2} in {(z + 1)}^{21} is (\begin{matrix} 21 \\ 2 \end{matrix})

z^{2} in z^{2} + 3 z + 3 is 1

Residue = (\begin{matrix} 21 \\ 2 \end{matrix}) + 1 = 210 + 1 = 211

Residue = 211

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