Given the exponential density function $f_X(x)=\lambda e^{-\lambda x}$, the nth moment is $\mu_n=n!/\lambda^n$.
If using the series representation to find the moment generating function $g(t)$:
$$g(t)=\sum_{k=0}^{\infty}\frac{\mu_kt^k}{k!}=\sum_{k=0}^{\infty}\left( \frac{t}{\lambda} \right)^k=\frac{\lambda}{\lambda-t}$$
Which only converges if $|t|<\lambda$. However, finding $g(t)$ from integration:
$$g(t)=\int_{0}^{\infty}e^{tx}\lambda e^{-\lambda x} = \left. \frac{\lambda e^{(t-\lambda)x}}{t-\lambda} \right|^\infty_0 =\frac{\lambda}{\lambda-t}$$
Which converges for $t < \lambda$ and appears to not have a lower bound. I'm wondering why there appears to be a difference in the region of $t$ in which the MGF converges for the same pdf, and which one is correct?