TITLE: At what point does exponential growth dominate polynomial growth?
QUESTION [6 upvotes]: It's well-known that exponential growth eventually overtakes polynomial growth (link, link). 
So for any non-negative integer $d$ and positive $\epsilon$, there exists $t^* \ge 0$ for which 
$$
1 + t + \frac{t^2}{2!} + \ldots + \frac{t^d}{d!} \le e^{\epsilon t} 
$$
for all $t \ge t^*$. 
In other words, it takes $t^*$ seconds for the exponential to catch up with the polynomial. 
I'd like to know if there's a closed-form expression for $t^*$ in terms of $d$ and $\epsilon$. Something like 
$$
t^* = d/\epsilon.
$$
Thoughts?

REPLY [2 votes]: It is enough to take $t^* = e(d+1)!/\epsilon^{d+1}$.
Proof:
The LHS of the desired inequality is at most $t^d/0! + t^d/1! + \cdots t^d/d! \le et^d$ (assuming $t\ge 1$ which is fine).
The RHS of the desired inequality is at least $(\epsilon t)^0/0! + \cdots + (\epsilon t)^{d+1}/(d+1)! \ge (\epsilon t)^{d+1}/(d+1)!$.
Therefore, it is enough if we ensure that $et^d \le (\epsilon t)^{d+1}/(d+1)!$. But this is equivalent to $t\ge e(d+1)!/\epsilon^{d+1}$.
With more effort one could presumably improve the dependence of $t^*$ on $d$ and $\epsilon$.