use \[{{1 \over 2}\over{r-1}} + {{1 \over 2}\over{r+1}} = {r \over{(r-1) (r+1)} }\] to sum the series
\[Sum_{n} = {2\over{1\times3}} - {4\over{3\times5}} + ... + {(-1)^{n-1} 2^n \over{(2n-1)(2n+1)}}\]

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tried allsorts here but not got anywhere, difference method etc, there must be something i'm missing

what is a good form

that is exactly as the question was written

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