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$$f(x)=\sum_{k=0}^\infty k^3(x-2)^k$$

I am supposed to find the $f(x)$ that this Taylor polynomial represents. How do I do this? I've tried using standard polynomials and I've tried differentiating those for $(1-x)^{-1}$ but I haven't been able to get a proper function out of it.

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    what says Wolfram alpha? – 2017-01-25
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    I tried to search for a function to revert Taylor expansions but I couldn't find one. The answer below seems promising so I'll try that one out. – 2017-01-25

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Try differentiating,

$$f(x)=\sum_{k=0}^{\infty} (x-2)^k$$

$$=\frac{1}{1-(x-2)}$$

If it converges.

Then multiply both sides by $x-2$ and repeat.

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    Trying that right now. Upon differentiating, should I increase the $k=0$ everytime by one or keep it at $0$? – 2017-01-25
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    By keeping $k=0$ I got to the correct answer of $f(x)=(x^2-3)\frac{x-2}{(x-3)^4}$. Thank you! – 2017-01-25