absolutely convergent integral and an additional question on asymptotics

Question

I want to show that the integral $$\int_1^{\infty}e^{-a(x-\ln(x))}\frac{x}{e^{cx}(1+x^2)}dx$$ is absolutely convergent by the dominated convergence theorem. So I figured that since $$e^{-a(x-\ln(x))}\frac{x}{e^{cx}(1+x^2)} \leq e^{-2bx}x^{a}$$ where $b=\max\{a,c\}$ one gets $$\int_1^{\infty}e^{-a(x-\ln(x))}\frac{x}{e^{cx}(1+x^2)}dx \leq \int_1^{\infty}e^{-2bx}x^{a}dx=2^{a-1}b^{-a-1} \Gamma(a+1,2b)<\infty$$ which completes the proof. Is that correct? In addition I want to find coefficients $g_k$ so that $$1-\frac{1}{t} \sim \sum_{k=0}^{\infty} (k+2)g_k(t-1)^k, \qquad{t \rightarrow 1}$$ My solution is $g_0=0$ and $g_k=\frac{(-1)^k}{k+2}$ for $k=1,2,3,\ldots$ Thanks in advance! :)