Is there a closed-form expression for this series?


$\displaystyle\sum\_{k\geq 1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}$


Any answers, ideas or references would be appreciated.



Closed form expression for this infinite series?
No closed form expression in terms of elementary functions, but if you are satisfied with special functions (incomplete Gamma function $\Gamma$ and exponential integral Ei), then
$$\displaystyle\sum\_{k=1}^\infty \frac{\lambda^k e^{-\lambda}}{k!}\cdot[1-(1-x)^k]\cdot \frac{1}{k}=$$
$$=e^{-\lambda}\,\Re \bigg(\text{Ei}\left(\lambda\right)+\ln (x-1)+\Gamma \left[0,\lambda (x-1)\right]\biggr),\;\;\lambda,x\in\mathbb{R}.$$
($\Re$ indicates the real part.) Here is a plot of the $x$-dependence for $\lambda=+1$ (blue) and $\lambda=-1$ (gold).


![](https://ilorentz.org/beenakker/MO/sumEiGamma.png)


