# Thread: Infinite summation of modified power series (casella, berger 2nd ed [formula 7.2.22])

1. ## Infinite summation of modified power series (casella, berger 2nd ed [formula 7.2.22])

Hi all,

Would really appreciate if someone could assist me on the following please: Casella and berger provide an example on pg 329 where they state:

$\displaystyle$-t_1+\log t_1\sum_{x_1=0}^{\infty}x_1\frac{e^{-t_1^{(r)}}(t_1^{(r)})^{x_1}}{x_1!}= -t_1 +t_1^{(r)}\log t_1$$I was having problems seeing how \displaystyle \sum_{x_1=0}^{\infty}x_1\frac{e^{-t_1^{(r)}}(t_1^{(r)})^{x_1}}{x_1!}$$ can therefore equal $\displaystyle$t_1^{(r)}$$. Please could someone help, thanks!! 2. ## Re: Infinite summation of modified power series (casella, berger 2nd ed [formula 7.2. Originally Posted by shotgun1 Hi all, Would really appreciate if someone could assist me on the following please: Casella and berger provide an example on pg 329 where they state: \displaystyle -t_1+\log t_1\sum_{x_1=0}^{\infty}x_1\frac{e^{-t_1^{(r)}}(t_1^{(r)})^{x_1}}{x_1!}= -t_1 +t_1^{(r)}\log t_1$$

I was having problems seeing how $\displaystyle$\sum_{x_1=0}^{\infty}x_1\frac{e^{-t_1^{(r)}}(t_1^{(r)})^{x_1}}{x_1!}$$can therefore equal \displaystyle t_1^{(r)}$$. Please could someone help, thanks!!
Writing the 'infinite sum' in a more 'standard form' You can verify that...

$\displaystyle \sum_{n=0}^{\infty} n\ \frac{e^{-t}\ t^{n}}{n!} = e^{-t}\ \sum_{n=1}^{\infty} \frac{t^{n}}{(n-1)!} = t\ e^{-t}\ \sum_{n=0}^{\infty} \frac{t^{n}}{n!}=t$

Marry Christmas from Serbia

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