Suma z symbolem Newtona

[Emiel Regis]

Zaćmienie mam i prosze mi to zsumować:
\(\displaystyle{ \sum_{n=k}^{\infty} p^k(1-p)^{n-k} \frac{\lambda^n}{n!}e^{-\lambda}}\)

Wynik:
\(\displaystyle{ \frac{(\lambda p)^k}{k!}e^{-\lambda p}}\)

[sigma_algebra1]

\(\displaystyle{ ...=\sum_{n=k}^{\infty}\frac{n!}{k! (n-k)!}p^k(1-p)^{n-k}\frac{\lambda^n}{n!}e^{-\lambda}=\frac{p^k}{k!}e^{-\lambda}{\lambda}^k
\sum_{n=k}^{\infty}\frac{[(1-p)\lambda]^{n-k}}{(n-k)!}=\frac{(p \lambda)^k}{k!}e^{-\lambda}e^{(1-p)\lambda}=\frac{(p \lambda)^k}{k!}e^{-p\lambda}}\)