(i) Let $X$ be a random variable. Use Markov's inequality to show that

$$\mathbb{P}(X \geqslant k) \leqslant \mathbb{E}\left(e^{t X}\right) e^{-k t}$$

for all $t \geqslant 0$ and real $k$.

(ii) Calculate $\mathbb{E}\left(e^{t X}\right)$ in the case where $X$ is a Poisson random variable with parameter $\lambda=1$. Using the inequality from part (i) with a suitable choice of $t$, prove that

$$\frac{1}{k !}+\frac{1}{(k+1) !}+\frac{1}{(k+2) !}+\ldots \leqslant\left(\frac{e}{k}\right)^{k}$$

for all $k \geqslant 1$.