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The sum to n terms of series 1+(1+1/2+1/...

The sum to `n` terms of series `1+(1+1/2+1/(2^2))+(1+1/2+1/(2^2)+1/(2^3))+` is

Text Solution

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The correct Answer is:
`2n-4/3(1-1/(2^(2n)))`
`S_(n)=1+(1-(1/2)^(3))/(1-1/2)+(1-(1/2)^(5))/(1-1/2)+..`n times
`=1+2(1-(1/2)^(3))+2(1-(1/2)^(5))+..` n times
`=1+(2+2+…`(n-1) times-`[(1/2)^(2)+(1/2)^(4)+..`(n-1) times`]`
`=1+2(n-1)-((1//2)^(2))/(1-(1/2)^(2))(1-(1/2)^(2n-2))`
`=2n-1-1/3+4/(3.2^(2n))`
`=2n-4/3(1-1/(2^(2n)))`
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