Need help solving this inequality involving factorial

I really have no idea how to solve this inequality, I would be happy with just a hint on how to solve it :)

$$\frac{e^{0.5}0.5^{n+1}}{(n+1)!} \le 10^{-6}$$

Thanks in advance!

Answer

I was going to suggest you to binary search the answer, but after computing the first 11 values I've got this:

• $f(0) = 0.8243606353500641$


• $f(1) = 0.20609015883751602$


• $f(2) = 0.03434835980625267$


• $f(3) = 0.004293544975781584$


• $f(4) = 0.0004293544975781584$


• $f(5) = 3.577954146484653\cdot10^{-05}$


• $f(6) = 2.5556815332033237\cdot10^{-06}$


• $f(7) = 1.5973009582520773\cdot10^{-07}$



• $f(8) = 8.87389421251154\cdot10^{-09}$


• $f(9) = 4.4369471062557704\cdot10^{-10}$


• $f(10) = 2.0167941392071684\cdot10^{-11}$

The value of n that suits you is 7.