Finding the exact value of a summation.

[NATURE.M]

Homework Statement

The sum we are given is Σ(from x=0->∞) [(x^2)(2^x)]/x!. We are asked to find the exact value of this sum using concepts discussed in class which include poisson random variables, and their expected values.

The Attempt at a Solution

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So i know the solution to the infinite sum is divergent. I know this since:

e^2 * Σ(from x=0->∞) (x^2) * Σ(from x=0->∞) (2^x)(e^-2)/x! = e^-2 * Σ(from x=0->∞) (x^2) since Σ(from x=0->∞) (2^x)(e^-2)/x! = 1 from poisson distribution. And we know Σ(from x=0->∞) (x^2) is divergent from simple infinite series. So the sum itself is ∞/or diverges.

But I don't know if there's a better way to solve this using only properties of the poisson distributions and their expected values/variances instead of invoking results from infinite series.

 

[Dick]

Homework Statement

The sum we are given is Σ(from x=0->∞) [(x^2)(2^x)]/x!. We are asked to find the exact value of this sum using concepts discussed in class which include poisson random variables, and their expected values.

The Attempt at a Solution

[/B]
So i know the solution to the infinite sum is divergent. I know this since:

e^2 * Σ(from x=0->∞) (x^2) * Σ(from x=0->∞) (2^x)(e^-2)/x! = e^-2 * Σ(from x=0->∞) (x^2) since Σ(from x=0->∞) (2^x)(e^-2)/x! = 1 from poisson distribution. And we know Σ(from x=0->∞) (x^2) is divergent from simple infinite series. So the sum itself is ∞/or diverges.

But I don't know if there's a better way to solve this using only properties of the poisson distributions and their expected values/variances instead of invoking results from infinite series.

The sum is not divergent. A ratio test will tell you that. I don't know how to sum it exactly, but it is convergent.

 

[NATURE.M]

The sum is not divergent. A ratio test will tell you that. I don't know how to sum it exactly, but it is convergent.

oh I think I noticed my error of breaking the sum up since its a property of finite sums only.

 

[LCKurtz]

Hint: If that sum had an $e^{-2}$ factor, what expected value of what distribution would it be calculating?

 

[NATURE.M]

Hint: If that sum had an $e^{-2}$ factor, what expected value of what distribution would it be calculating?

Yeah I kinda understand that what I'm uncertain about is how does Σ(from x=0->∞) (x^2)(2^x)(e^-2)/x! = 6. The x^2 confuses me ?

 

[LCKurtz]

Hint: If that sum had an $e^{-2}$ factor, what expected value of what distribution would it be calculating?

Yeah I kinda understand that what I'm uncertain about is how does Σ(from x=0->∞) (x^2)(2^x)(e^-2)/x! = 6. The x^2 confuses me ?

If you would answer my question it might help you.

 

[lurflurf]

use homogeneous differentiation for these types of exercises

$$\sum_{x=0}^\infty \dfrac{x^22^x}{x!}=\left. \left(r\dfrac{\mathrm{d}\phantom{r}}{\mathrm{d}r}\right)^2\sum_{x=0}^\infty \dfrac{r^x}{x!}\right|_{r=2}$$

or use
$$\sum_{x=0}^\infty \dfrac{r^x}{x!}
=\sum_{x=0}^\infty x\dfrac{r^{x-1}}{x!}
=\sum_{x=0}^\infty x(x-1)\dfrac{r^{x-2}}{x!}=e^x$$