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The sum of the series 1 +3/4+7/16 +15/64...

The sum of the series `1 +3/4+7/16 +15/64+31/256+....` to infinite is:

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`=4sum_(n=1)^oo 2^n/4^n-4sum_(n=1)^oo1/4^n`
`4sum_(n=1)^oo 1/2^n-4sum_(n=1)^oo 1/4^n`
`4[sum_(n=1)^oo 1/2^n-sum_(n=1)^oo 1/4^n]`
`4[(1/2)/(1-1/2)-(1/4)/(1-1/4)]`
`4[(1/2)/(1/2)-(1/4)/(3/4)]`
`4[1-1/3]=4*2/3=8/3`.
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Knowledge Check

  • Find the sum of the series 1/4-3/16+9/24-27/256+….

    A
    (1/3)
    B
    (1/4)
    C
    (1/7)
    D
    none of these

  • The sum of n terms of the series 1/2 +3/4 +7/8+15/16+.... is

    A
    `n-1+(1/2)^(n)`
    B
    1
    C
    n - 1
    D
    `1+2^(-n)`

  • Sum of the series 1 + 3 + 7 + 15 + 31 +… to n terms is

    A
    `2^(n) - 2 - n`
    B
    `2^(n + 1) + 2 + n`
    C
    `2^(n + 1) - 2 - n`
    D
    none

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