How do I show that the following improper integral $$\int^1_0\frac{e^x-1-x}{x^2\sqrt{x}}dx$$ is convergent by using Taylor expansion rigorously?
So $e^x=1+x+x^2/2+x^3B(x)$ where $B(x)$ bounded close to zero. It doesn't make sense to just insert this in the integral directly. But maybe the following works? There exists a $\delta>0$ such that $\int^\delta_0\frac{1+x+x^2/2+x^3B(x)-1-x}{x^2\sqrt{x}}dx=\int^\delta_0\frac{1/2}{\sqrt{x}}+\sqrt{x}B(x)dx$ which is convergent since $B(x)$ is bounded.