$$\sum_{n>=j}^{\infty }\frac{(1-\beta)^{n-j}*\lambda^n}{(n-j)!}$$
$j$ is a constant positive integer and $0<\beta<1$ also a constant.
This series converges, but I want to know the convergent value
If can, how to do that? or find the approximate value.
Thank you so much.
Let $m = n - j$. Then $m$ ranges from $0$ to $\infty$ as $n$ ranges from $j$ to $\infty$. Thus
$$\sum_{n = j}^\infty \frac{(1 - \beta)^{n-j}\lambda^n}{(n-j)!} = \sum_{m = 0}^\infty \frac{(1 - \beta)^m\lambda^{m+j}}{m!} = \lambda^j \sum_{m = 0}^\infty \frac{[(1 - \beta)\lambda]^m}{m!} = \lambda^j\exp\{(1 - \beta)\lambda\}.$$