Question

f(x) = x/(x^2 - 2x +2)
Obtain the power series representation of f in x-1

Answer 1

http://www.wolframalpha.com/input/?i=expand+x%2F%28x%5E2+-+2x+%2B2%29+at+x%3D1&dataset=

Answer 2

you might start with factorizing the denominator (both has complex roots) and then making separate fractions (partial fractions)
and then expand using geometric series for 1/((x-1) + something) ...
good luck.

Answer 3

Write it like
\[
\frac{x}{x^2-2
x+2}=
\\
\frac{x}{x^2-2
x+1+1}=\\
\frac{x}{(x-1)^2+1}=\\
\frac{x-1+1}{(x-1)^2+1}=\\
\frac{x-1}{(x-1)^2+1}+\frac{1}
{(x-1)^2+1}

\]
Use geometric series to expand the last line

Answer 4

\[
\Large
\frac 1{1+u}=\sum_{n=0}^\infty (-1)^n u^n\\
\Large
\frac 1{1+u^2}=\sum_{n=0}^\infty (-1)^n u^{2n}\\
\Large
\frac 1{1+(x-1)^2}=\sum_{n=0}^\infty (-1)^n (x-1)^{2n}\\

\Large
\frac {x-1}{1+(x-1)^2}=\sum_{n=0}^\infty (-1)^n (x-1)^{2n+1}\\

\]
Convergence is for
\[
|x-1|<1
\]

Answer 5

Combining, you obtain
\[
\frac{x}{x^2-2
x+2}=1+(x-1)-(x-1)^2-(x-1)^3+(x-1)^4+(x-1)^5-(x-1)^6-\\(x-1)^7+(x-1)^8+(
x-1)^9+O\left((x-1)^{10}\right)
\]

Answer 6

@experimentX

Answer 7

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