I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$
Can anybody help me?
sequences-and-series
asked 2011-09-25
user id:16653
31
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I tried $\LaTeX$-ing your question, but I'm not sure this is what you meant. What is $k$? – 2011-09-25
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You already have a few answers using the derivative of geometric series; another possible approach would be [summation by parts](http://en.wikipedia.org/wiki/Summation_by_parts). – 2011-09-25
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Note that you can check the answers given bellow at wolframalpha: http://wolframalpha.com/input/?i=sum%28n%2Ax%5En%29 – 2011-09-25
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Yeah, you defintly have to provide some information about k, since that sum may actually diverge, for instance, if k was 1. – 2011-09-25
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Yes, that's right. Thank you very much for the LATEXing. At this moment k is a constant with absolute value less than 1. But I am also curious about what results if k is larger than 1. – 2011-09-25
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Suppose $k=1/6$. Think about the probability that one throws a die and fails to get a "1" the first $n-1$ times and then gets a "1" the $n$th time. That is the probability that the number of throws needed to get a "1" is $n$. Then ask what is the _average_ number of throws needed to get a "1". The answer is precisely this sum. If you ask anyone---ranging from someone who never thinks about math to the most able mathematician---what they would guess is the average number of throws needed to get a "1", they will instantly say 6. And that is correct. – 2011-09-25
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