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
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 theres 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.