How do you find a power series representation for $f(x)= 9/(1-3x)^2$ and what is the radius of convergence?

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Answer 1 · 2018-02-16T19:57:12

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$3/(1-3x)^2 = d/(dx)(1/(1-3x))$ but for $abs (3x) < 1$

$1/(1-3x) = lim_(n->oo)sum_(k=0)^n3^kx^k$ hence

$f(x) = 9/(1-3x)^2 = 3sum_(k=0)^oo 3^kx^k$

convergent for $absx < 1/3$

How do you find a power series representation for f(x)= 9/(1-3x)^2 f(x)= 9/(1-3x)^2 and what is the radius of convergence?