My Math Forum Proof of tricky infinite series

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 June 10th, 2012, 10:40 AM #1 Newbie   Joined: Jun 2012 Posts: 2 Thanks: 0 Proof of tricky infinite series Salutations all. This question involves proving infinite series with differentiation, but I'm posting here because it's a proof question and so a harder question than just standard calculus which doesn't usually have proofs. I just have one issue with this question. I've been scratching my head out on how the solution got, deduced, or inferred the expression in the red box. Where on earth does it come from? In part (i), we proved $\sum\limits_{r=0}^{\infty }{\left( \begin{matrix} 2r \\ r \\ \end{matrix} \right){{\left( \frac{1}{8} \right)}^{n}}={{\left( 1-\frac{1}{2} \right)}^{-1/2}}}$ But how on earth can you get a general expression for $x$ from this? I'm missing something?!!!?!? http://i47.tinypic.com/27ybvux.png
 June 10th, 2012, 10:41 AM #2 Newbie   Joined: Jun 2012 Posts: 2 Thanks: 0 Re: Proof of tricky infinite series Salutations again. Sorry, it doesn't look like I can post pictures. The link to my question is at the bottom of my post. Thank you all!

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### tricky infiNite series question

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