Limit ratio test take derivative with respect to what? Suppose I define a sequence as follows: $a_n = a_{n-1} + n^2$.
Let's say I'd like to see if this sequence converges by performing the ratio test to see if $\frac{a_{n+1}}{a_n} < 1$.
Let's pretend that $\frac{a_{n+1}}{a_n}$ is either $\frac{\infty}{\infty} or \frac{0}{0}$ (I realize it's not) and requires applying l'Hopital's Rule.
In this case, what would I take the derivative with respect to, $a_n$ or to $n$? So would the numerator become $\frac{\partial}{\partial a_n}=1$ or $\frac{\partial}{\partial n}= 2n$?
 A: 
Let's say I'd like to see if this sequence converges by performing the ratio test to see if $\frac{a_{n+1}}{a_n} < 1$.

I assume you want to check the converges of the series $\sum a_n$ by applying the ratio test?
It would be the derivative with respect to $n$, the variable of $a_n = a(n)$. 

Suppose I define a sequence as follows: $a_n = a_{n-1} + n^2$.

Note that your example is a recursively defined sequence; it would be easier to work with an explicit formula to directly apply the ratio test. 
In any case, if you have expressions for $a_{n+1}$ and $a_n$ and the limit
$$\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_n}\right|$$
would give an indeterminate form where l'Hôpital applies ($\tfrac{0}{0}$ or $\tfrac{\infty}{\infty}$), then you can proceed by taking the derivative with respect to $n$.

To compare: in the context of real functions where you have a similar indeterminate form for a limit of the form $\tfrac{f(x)}{g(x)}$, you don't take the derivative with respect to $f(x)$ (and/or $g(x)$), but with respect to (the variable) $x$!
