Here are my difficulties, is there any method to solve or estimate the following integral?
$$\int_0^\infty\left[e^{-2λx}+(1-p) e^{-λx}-1\right]^n dx$$
where $\lambda$ and $p$ are constants.
following is my try:
let $Z=e^{-\lambda x}$, therefor we can write:
$$f(Z)=[Z^2+(1-p)Z-1]^n $$
tailor expansion of $f(Z)$ for the first two terms is as follows:
$$f(0)=[-1]^n$$ $$f'(Z)=n[Z^2+(1-P)Z-1]^{n-1}(2Z+(1-p)) $$ $$f'(0)=n[-1]^{n-1}(1-p) $$
and the tailor expansion is as follows:
$$f(Z)\approx f(0)+f'(0)\times Z$$ $$f(x)\approx [-1]^n+n[-1]^{n-1}(1-p)\times e^{-\lambda x}$$
final estimation is : $$\int_0^\infty\left[[-1]^n+n[-1]^{n-1}(1-p)\times e^{-\lambda x}\right]dx$$
the problem is with the $[-1]^n$