The integral we're dealing with, is $\int ^\infty_0 \frac{1-e^x}{x^p} dx$
I've a couple of questions, apart from how to determine the values of $p$ for which the integral converges -
Is it okay to use the Taylor expansion of $e^x$ here? Why, or why not? If the expansion is used, what we end up with is an infinite series which can integrated (at least in an indefinite manner).
When is it alright to switch integrals and sums, especially when we're dealing with improper integrals and infinite sums?
Is (1) the right approach for this problem? (feels like the integral converges for no value of $p$, if I'm right)
Please help me out here, any and all help is appreciated.