\displaystyle EY = \sum_{k=0}^{\infty}\frac{1}{1+k}P(Y=\frac{1}{1+k}) = \sum_{k=0}^{\infty}\frac{1}{1+k}P(X=k) = \sum_{k=0}^{\infty}\frac{1}{1+k}\frac{\lambda^k}{k!}e^{-\lambda} = e^{-\lambda}\sum_{k=0}^{\infty}\frac{\lambda^k}{(k+1)!} = \frac{e^{-\lambda}}{\lambda}\sum_{k=0}^{\infty}\frac{\lambda^{k+1}}{(k+1)!} = \frac{e^{-\lambda}}{\lambda}(e^{\lambda}-1) = \frac{1-e^{-\lambda}}{\lambda}