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Math Help - find the sume of a series

  1. #1
    Junior Member
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    find the sume of a series

    Find the some of S = \sum_{i = 1}^{\infty} (\frac{1}{i + 1} - \frac{1}{i + 2})
    plugging index values in.

    (\frac{1}{2} - \frac{1}{3})  + (\frac{1}{3} -\frac{1}{4} ) + (\frac{1}{4} -\frac{1}{5} )

    I note that by regrouping the terms cancel out.

    \frac{1}{2}  + (- \frac{1}{3}  + \frac{1}{3} ) + (-\frac{1}{4}  + \frac{1}{4} ) -\frac{1}{5} . . .

    and looks like the sum is 1/2 but I still have the tail and I have not show how it goes to 0 as n goes to infinity. how would i phrase the tail to get a limit resulting in zero?
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  2. #2
    Senior Member Spec's Avatar
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    = \frac{1}{2}+\left(-\frac{1}{3} + \frac{1}{3}\right)+...+\left(-\frac{1}{n-1} + \frac{1}{n-1}\right) - \frac{1}{n}=\frac{1}{2}+\frac{1}{n}=\frac{1}{2} when n \to \infty
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