(Calc II) Direct Comparison Test For Infinite Series When comparing two functions, say $n^4$ and $n^3+20$, is it enough (for the purposes of the direct comparison test) to say that $n^4 > n^3+20$ near infinity, or do I have to find a function that is larger than $n^3+20$ for every $n$?
Thanks in advance. :)
 A: For all of these convergence theorems, the key notion is eventually. It's enough that $n^4 > n^3 + 20$ eventually. This is intuitively clear if you're familiar with end behavior of polynomials. But if you're trying to be rigorous, you have to demonstrate that there is some index $N$ such that the inequality $n^4 > n^3 + 20$ holds for all $n > N$.
Such an $N$ isn't too hard to find by an exploration that I call "wishful thinking". We wish that $n^4 > n^3 + 20$. What would guarantee this to be true? If $n^3 > 20$ (so $n > 2$ works), then
$$
2n^3 = n^3 + n^3 > n^3 + 20. 
$$
But when is $n^4 > 2n^3$? Once $n > 2$, since
$$
n^4 = n \cdot n^3 > 2 \cdot n^3. 
$$
I'm using the fact that these polynomial quantities are all positive throughout.
Putting all this together. For $n > 2$, we have
$$
n^4 > 2n^3 > n^3 + 20, 
$$
as desired.
Incidentally, the positive root of $f(x) = x^4 - (x^3 + 20)$ (where the curves intersect) is at $x \approx 2.417\dots\,$, which you can see on the graph, so among natural numbers the inequality holds beginning with $n=3$.
