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Find the sum to ininity of the series 1-...

Find the sum to ininity of the series `1-3x+5x^2+7x^3+..... oo` when Ix| <1.

Text Solution

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Let `S_(oo)` denotes the sum of the given infinite series. Now,
`S_(oo)=1-3x+5x^(2)-7x^(3)+..oo`
`=1+3(-x)+5(-x)^(2)+7(-x)^(3)+..oo`
Here, a=1,r=-x and d=2. Hence,
`S_(oo)=1/(1-(-x))+(2(-x))/([1-(-x)]^(2))`
`=1/(1+x)-(2x)/((1+x)^(2))`
`=(1-x)/((1+x)^(2))`
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