The integral $\displaystyle \int_{0}^{\infty} \Big( \frac{x^{a-1}}{\sinh x} - x^{a-2} \Big) \ dx$ converges for the range $0 \le a < 1$. The analysis utilizes the series expansion of $\frac{1}{\sinh x}$, leading to the simplification $\frac{x^{a-1}}{\sinh x} - x^{a-2} \sim - \frac{x^{a}}{6}$ as $x \to 0$ and $\sim - x^{a-2}$ as $x \to \infty$. The convergence is confirmed by evaluating the behavior of the integral at both limits, establishing that it converges for $-1 < a < 1$.
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Random Variable said:How do you show that $\displaystyle \int_{0}^{\infty} \Big( \frac{x^{a-1}}{\sinh x} - x^{a-2} \Big) \ dx $ converges for $0 \le a <1$?