(Cinlar, probability and stochastic, p.345) $X$ is Levy process in $\mathbb R^d$. Suppose that $\mathbb P(|X_t| > (1+b)n) \le e^t c^n$ for $n \in \mathbb N$, $c \in (0,1)$, and $b \in \mathbb R_{+}$. I want to verify that every moment of $X_t$ is finite.
To do this, for $k \in \mathbb N$, $$\mathbb E |X_t|^k = \int_0^\infty k x^{k-1} \mathbb P(|X_t|>x) dx \le \sum_{j=0}^\infty \int_{j(1+b)}^{(j+1)(1+b)} k x^{k-1} \mathbb P(|X_t|> j(1+b)) = e^t (1+b)^k\sum_{j=0}^\infty c^j[(j+1)^k-j^k].$$ $\sum_{j=0}^\infty c^j[(j+1)^k - j^k] \le \sum_{j=0}^\infty[(j+1)^k - j^k]$ since $c \in (0,1)$, but isn't the last term zero? This seems weird, and I don't know where I make a mistake. I can't think of another route. Can you give me some help or any suggestion?
The bound $c^j\left((j+1)^k-j^k\right)\leqslant (j+1)^k-j^k$ is too crude, since the series $\sum_j ((j+1)^k-j^k)$ diverges. Nevertheless, you can simply say that $0\leqslant c^j\left((j+1)^k-j^k\right)\leqslant c^j(j+1)^k$ and the series $\sum_{j=0}^\infty c^j(j+1)^k$ converges by the ratio test.