# Series Question (from a probability question)

1. Sep 17, 2007

### mattmns

[SOLVED] Series Question (from a probability question)

EDIT: Found it, never mind.

Here is my question:

How do I find what $$\sum_{k=1}^{\infty}k(1/2)^k$$ is?

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Here is the original question and my work.
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A couple decides to continue to have children until a daughter is born. What is the expected number of children of this couple?
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So suppose P(having a daughter) = 1/2.

Define the random variable X = number of children until a daughter is born.

Then $f_X(x) = (1-1/2)^{x-1}(1/2)$

So,
$$E(X) = \sum_{x = 1}^{\infty}x f_X(x) = \sum_{k=1}^{\infty}k(1/2)(1/2)^{k-1}$$

or as I wrote it above.

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Now I believe everything I have done up this point is correct, however I can't remember how to do series. I have been looking at my calculus book, but I have yet to find what I need. Any hints would be greatly appreciated, thanks!

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EDIT: Found it online, seems to be a common geometric series, not sure why I couldn't find it in my book.

Last edited: Sep 17, 2007
2. Sep 17, 2007

### Dick

You know how to find f(x)=sum(x^n), right? f'(x)=sum(n*x^(n-1)). Is that enough of a hint?

3. Sep 17, 2007

### Dick

It's good to know how it's derived as well.

4. Sep 17, 2007

### mattmns

Thanks, indeed it is good to know how to derive it.