@SillyGoose Causal Perturbation theory is good if you can understand it

@Obliv That is the $\beta\to\infty$ and $\beta\to0$ limits. You know the Taylor expansion of exp, which applies to the zero limit. The infinite limit, convert to negative exponents, so that then the exp goes to zero.

@SillyGoose ok, nice; when you are doing these integrals, say the last one, is it $\int_0^{\pi/2}\cos^4\theta_{n+1}\,\mathrm d\theta_{n+1}$ alone, or are there a lot of $\int_0^{\pi/2}\mathrm d\theta_i$ to also integrate?

Anyway, I'm sure you will find the following identity useful $$\cos^2\theta=\frac{1+\cos2\theta}2\qquad\bigwedge\qquad\sin^2\theta=\frac{1-\cos2\theta}2$$

m i a o ~

thanks, I did have to give up and look at the solution. I'm definitely memorizing it I've seen it this many times

power series, geometric series, etc. I'm going to do a nice review of fundamentals of calculus after finals