I would like to know how to prove $$e^\pi-\pi\sim 20.$$ More precisely, I want to show by using only mathematical tools that, $$19.999<e^\pi-\pi<20$$
I have checked with online calculator and I got $$e^\pi-\pi\approx19.9990999792\sim 20.$$
I tried to use the Tyalor expansion for exponential $$ e^\pi =\sum_{n=0}^{\infty} \frac{\pi^n}{n!} = \pi +1+\sum_{n=2}^{\infty} \frac{\pi^n}{n!}$$ then, $$e^\pi -\pi =1+\sum_{n=2}^{\infty} \frac{\pi^n}{n!}$$ which is not easy to continue from here, since the factor $\pi^n$ is involved. Any idea?