How to calculate $\sum_{k=0}^\infty e^{-3} \frac{k}{k+1} \frac{3^k}{k!} $.

Question

I've been dealing probability questions lately, and I had to calculate this sum : $\sum_{k=0}^\infty e^{-3} \frac{k}{k+1} \frac{3^k}{k!} $ , which I have no idea how to bring to to an easier form, or for example extract from it something that is known. Its good to mention that its a result of multiplying with Poisson function of probability with parameter $3$.

Answer 1

Note that if we let $k=n-1$,

$$\frac k{k+1}\frac{3^k}{k!}=\frac{k3^k}{(k+1)!}=\frac13\frac{(n-1)3^n}{n!}=\frac13\left(\frac{3^n}{(n-1)!}-\frac{3^n}{n!}\right)$$

Summing from $n=1$ to infinity yields

$$3e^3S=\sum_{n=1}^\infty\left(\frac{3^n}{(n-1)!}-\frac{3^n}{n!}\right)=3e^3-e^3+1=1+2e^3$$

Thus,

$$\sum_{k=0}^\infty e^{-3}\frac k{k+1}\frac{3^k}{k!}=\frac{1+2e^3}{3e^3}$$