`sum_(n=0)^oo(-1)^nx^n`

`=sum_(n=0)^oo(-1x)^n`

It is a geometric series with common ratio`r=-x` ,so the series converges for `|r|<1`

`|-x|<1`

`=>|x|<1`

`=>-1<x<1`

In this case, Sum of the series=`a/(1-r)`

`=1/(1-(-x))`

`=1/(1+x)`

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