I am trying to calculate this sum:
$\sum_{k=1}^\infty\frac{q\lambda^k}{k!e^\lambda}$
My solution is:
$\begin{align*} \sum_{k=1}^\infty\frac{q\lambda^k}{k!e^\lambda}&=\frac{q}{e^\lambda}\sum_{k=1}^\infty\frac{\lambda^k}{k!}=\frac{q}{e^\lambda}\sum_{k=0}^\infty\frac{\lambda^{k+1}}{(k+1)!}\\ &=\frac{q\lambda}{e^\lambda\cdot (k+1)}\sum_{k=0}^\infty\frac{\lambda^k}{k!}\lambda=\frac{q\lambda}{e^\lambda\cdot (k+1)}\cdot e^\lambda=\frac{q\lambda}{k+1}\end{align*}$
But the solution given by my tutor is $q(1-e^{-\lambda})$. Could someone please verify my calculation and tell me where I went wrong?