Sum of series, x >= 0, [ x (n*theta)^x ] / [ (e^{x/n} - 1) x! ]

[Marina5]

Hi!
Could you help me, please, calculate

$\displaystyle \displaystyle\sum_{x \geq 0} \frac{x\left(n\theta\right)^x}{\left(e^{\frac{x}{n}}-1\right)x!}$

I know it converges.

Thank you!

 

[mmm4444bot]

Hello Marina. Do you have any beginning work to share? Where in this exercise do you need help?

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[blamocur]

Hi!
Could you help me, please, calculate
$\displaystyle \displaystyle\sum_{x \geq 0} \frac{x\left(n\theta\right)^x}{\left(e^{\frac{x}{n}}-1\right)x!}$
I know it converges.
Thank you!

Are you summing by $x$ or by $n$ ?

 

[Marina5]

Are you summing by $x$ or by $n$ ?

by x, n and theta are constants

 

[blamocur]

by x, n and theta are constants

You mean $\sum_{x=0}^\infty ...$ with integer $x$? Is $n$ an integer or an arbitrary real number ?
Also, what have you tried and which problem have you encountered?

 

[BigBeachBanana]

Looks like the expectation of some conditional Posterior Distribution to find the marginal probability
$$\Pr(X=x) = \sum x \cdot \Pr(X=x|\theta,n)$$