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Math Help - summation problem

  1. #1
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    summation problem

    The problem is the summation of (n/(n+1)!) from n=1 to infinity.
    any help that can be given is greatly appreciated!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Is...

    \displaystyle \frac{n}{(n+1)!}= \frac{n+1}{(n+1)!} - \frac{1}{(n+1)!}= \frac{1}{n!} - \frac{1}{(n+1)!} (1)

    ... so that...

    \displaystyle \sum_{n=1}^{\infty} \frac{n}{(n+1)!} =  \sum_{n=1}^{\infty} \frac{1}{n!} - \sum_{n=1}^{\infty} \frac{1}{(n+1)!} = \sum_{n=1}^{\infty} \frac{1}{n!} - \sum_{n=2}^{\infty} \frac{1}{n!} = 1 (2)

    Kind regards

    \chi \sigma
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  3. #3
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    sorry I have not done a summation in a long time but how do you know that the summation of 1/n! from one to infinity minus the summation of 1/n! from 2 to infinity is 1?
    thanks so much for the help
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  4. #4
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    Quote Originally Posted by casey View Post
    sorry I have not done a summation in a long time but how do you know that the summation of 1/n! from one to infinity minus the summation of 1/n! from 2 to infinity is 1?
    thanks so much for the help
    Write out the first few terms and simplify.
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