# Does $\sum_{k = 0}^{\infty} \sum_{n = 0}^{\infty}\frac{B^k C^{(n+k+1)}}{(ib)^n k! (n+k+1)!}$ converge?

In relation to my question: Finding the residue of function with Laurent series $\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{y^n(A+By+Cy^{-1})^k}{\beta (\beta i)^n \ k!}$ I need to find an expression for $\sum_{k = 0}^{\infty} \sum_{n = 0}^{\infty}\frac{B^k C^{(n+k+1)}}{(ib)^n k! (n+k+1)!}$, but I'm stuck, I don't even know if it converges. Any help, hints or suggestions would be much appreciated

I have the answer now :) The sum above can be rewritten as \begin{align}e^B \sum_{n=0}^{\infty} \sum_{k=0}^{n}\frac{C^{n+1}}{(i b)^k(n+1)!} &= e^B \sum_{n=0}^{\infty} \frac{C^{n+1}(1-(1/ib)^{n+1})}{(1+(i/b))(n+1)!}\\ &= e^B\left(\frac{e^C-e^{\frac{-iC}{b}}}{1+\frac{i}{b}} \right) \end{align} Since $\sum_{k=0}^{n}\frac{1}{(i b)^k} = \frac{1-(1/ib)^{n+1}}{1+(i/b)}$, for $b \neq -i$