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I have been asked to expand (1-2t)^(-n/2). So far I have found the log of this function and then differentiated twice in order to find my mean and variance as n and 2n respectively. How do I write my final answer as a power series in t as far as the term in t^2?

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    That would be great! Thank you. =)2010-12-08

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You have already done most of the work. If the mean \mu'_1 and variance $\sigma^2$ are equal to $n$ and $2n$, respectively, then the second moment \mu'_2 is given by \mu'_2 = \sigma^2 + (\mu'_1)^2 = n(n+2). Now, the moment-generating function can be expanded as follows: M(t) = 1 + \mu' _1 t + \frac{1}{{2!}}\mu' _2 t^2 + \cdots = 1 + nt + \frac{{n(n + 2)}}{2}t^2 \cdots.

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If you want to expand your function in a power series on the argument $t$ you can write its Taylor Polynomials .