Question

If \( f(x) = O(x^r) \) as \(x \rightarrow \infty \), show that\[\int_0^{f(x)}{e^{-u}(1+\frac{u}{x})^x}du = \int_0^{f(x)}\exp(\frac{-u^2}{2x}+\frac{u^3}{3x^2}-...+\frac{(-1)^{m-1}u^m}{mx^{m-1}})}du + O(x^{-s})\]

Answer 1

\[\int_0^{f(x)}{e^{-u}(1+\frac{u}{x})^x}du = \int_0^{f(x)}\exp(\frac{-u^2}{2x}+\frac{u^3}{3x^2}-...+\frac{(-1)^{m-1}u^m}{mx^{m-1}})du + O(x^{-s})\]