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Does the series k a^k have a name?

  1. May 11, 2013 #1
    Hi, a naive question here, but I was wondering if the series

    [itex]\sum^{∞}_{k=0}[/itex] k a[itex]^{k}[/itex]

    has a particular name? As in 'geometric series' for [itex]\sum^{∞}_{k=0}[/itex] a[itex]^{k}[/itex] ?

    And what about the more general [itex]\sum^{∞}_{k=0}[/itex] k[itex]^{n}[/itex] a[itex]^{k}[/itex] ?


    As a related question, you seem to be able to get the formula for the sum by differentiating the formula for the geometric series. Is this a correct and mathematically rigorous way to do it?

    Cheers,
    zeroseven
     
  2. jcsd
  3. May 11, 2013 #2

    tiny-tim

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    hi zeroseven! :smile:

    no, no names so far as i know
    yes, so long as both limits exist, d/dx ∑ fn(x) = ∑ d/dx fn(x) is rigorous

    (can you prove it? :wink:)
     
  4. May 11, 2013 #3
    Thanks tiny-tim!
    I thought that might be the case. So many other series seem to have been named that I just thought it was odd!

    No, I can't prove the identity you gave in your post... yet! But I will try.
     
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