How do you find a power series representation for $(x/(2-x))^3$ and what is the radius of convergence?

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Answer 1 · 2016-10-21T17:25:16

$(x/(2-x))^3 equiv sum_(k=2)^oo(k(k-1))/2^(k+2)x^(k+1)$ which is convergent for $abs(x) < 2$

We have $(x/(2-x))^3=(x/2)^3(1/(1-x/2))^3$

Calling $y = x/2$ we have

$(x/(2-x))^3 equiv y^3(1/(1-y))^3$

but

$d^2/dy^2(1/(1-y))=2(1/(1-y))^3$

Now, for $abs y < 1$ we have

$1/(1-y)=sum_(k=0)^ooy^k$ then

$y^3(1/(1-y))^3 = y^3/2d^2/dy^2 sum_(k=0)^ooy^k=y^3/2 sum_(k=2)^ook(k-1)y^(k-2)$ or

$y^3(1/(1-y))^3=1/2sum_(k=2)^ook(k-1)y^(k+1)$

but $y=x/2$ so finally

$(x/(2-x))^3 equiv sum_(k=2)^oo(k(k-1))/2^(k+2)x^(k+1)$ which is convergent for $abs(x/2) < 1$ or $abs(x) < 2$

How do you find a power series representation for (x/(2-x))^3 (x/(2-x))^3 and what is the radius of convergence?