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Geometric distribution , cumulative from characteristic function watch

1. 1. The problem statement, all variables and given/known data

Hi, I have the probabilty density:

and I am asked to find the characteristic function:

and then use this to determine the mean and variance of the distribution.

2. Relevant equations

I have the general expression for the characteristic function :

* ,
from which can equate coefficients of to find the moments.

3. The attempt at a solution

So I have

I understand the solution given in my notes which is that this is equal to, after some rearranging etc, expanding out using taylor :

and then equating coefficients according to * However my method was to do the following , and I'm unsure why it is wrong:

And so comparing to *

Anyone tell me what I've done wrong? thank you, greatly appreciated.
2. (Original post by xfootiecrazeesarax)

I don't understand why the 2nd equality holds here (I am doubtful that it does...)
3. (Original post by DFranklin)
I don't understand why the 2nd equality holds here (I am doubtful that it does...)
expanded my exponential incorrectly, falling alseep, thank you !

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Updated: January 11, 2017
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