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Infinite Summations

  1. May 26, 2008 #1
    Looking for ways to solve infinite summations, I found an ancient topic here talking about solving infinite summations that come out to answers with pi.
    How would I solve an infinite summation that does not come out to an answer with pi?

    Such as:
    [tex]\sum_{n=1}^{\infty}\frac{n+1}{6^n} [/tex]

    The solution is 11/25, btw.
    My attempt: I am not really experienced with this area of math, so what I did was put it into my TI-nSpire but it couldn't do it because it's not the CAS version. I plugged in a large number such as 999 terms instead of infinity terms and it came out to the right answer, but I am looking for a more "correct" way to solve the problem. I also thought of finding the sum of a geometric sequence but I realized that doesn't really work for most summations.

    Oh, I'm also new to these forums, so hi to everybody! =D

    EDIT: The n under the summation should say n=1
    Last edited: May 26, 2008
  2. jcsd
  3. May 26, 2008 #2


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    Many of these problems can be solved by recognizing that your series is actually a special case of a more general one.

    One of the more useful series is the geometric series: If |x|<1,
    [tex]\sum_{n=0}^{\infty} x^n = \frac{1}{1-x}.[/tex]

    It's not immediately obvious how we can use this to evaluate your series. However, there's a nice trick: Differentiate both sides of the above equation term by term to get (for |x| < 1):
    [tex]\sum_{n=1}^{\infty} n x^{n-1} = \frac{1}{(1-x)^2}.[/tex]

    Can you take it from here?
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