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For all $x\in(-1,1)$, the following series converges $$\sum_{n=0}^{\infty}(-1)^{n}(n-2) x^{n}$$ and defines a function $f:(-1,1)\rightarrow\mathbb{R}$. Find a closed form for $f(x)$, $x\in(-1,1)$.

Appreciating any help, and any methods to solve such things in general.

Blue
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Rack Cloud
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  • and any methods to solve such things in general – Rack Cloud Oct 08 '19 at 22:36
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    $f(x)/x^3$ has the general term $(-1)^n(n-2)x^{n-3}$ which is the derivative of $(-1)^n x^{n-2}$. This series you can sum in closed form (geometrical series) and then differentiate. – Winther Oct 08 '19 at 22:40
  • You should include something of what you know about this problem. (What have you tried? Where did you get stuck? etc) This information helps answerers tailor their responses to best serve you, without wasting time (theirs or yours) telling you things you already know or using techniques with which you are unfamiliar. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Edit your question to add any details; comments are easily overlooked. – Blue Oct 08 '19 at 22:42
  • trying to learn this for an exam it's an old exam – Rack Cloud Oct 08 '19 at 22:48
  • $$ f(x) / x^{3} \text { has the general term }(-1)^{n}(n-2) x^{n-3} \text { which is the derivative of }(-1)^{n} x^{n-2} $$ first what is f(x) second how did you find x^3 – Rack Cloud Oct 08 '19 at 22:50
  • This might be useful – Bumblebee Oct 08 '19 at 23:15
  • don't understand that – Rack Cloud Oct 08 '19 at 23:21

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