<s>[INST] Prove inquality involving factorial This is part of an analysis problem I'm working on.
Show that:
$ \frac{1}{(n+1)!} + \frac{1}{(n+2)!} + ... < \frac{3}{(n+1)!}$
After some algebra I got
$ \frac{1}{n+2} + \frac{1}{(n+2)(n+3)} + \frac{1}{(n+2)(n+3)(n+4)} + ... < 2$
Now I'm stuck.
 [/INST] If $n \geq 0$, you have
$$ \frac{1}{n+2} + \frac{1}{(n+2)(n+3)} + \frac{1}{(n+2)(n+3)(n+4)} + \cdots$$
$$ \leq
  \frac{1}{2} + \frac{1}{2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \cdots $$
$$ < \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \cdots = 1 < 2$$
</s>