Techniques for Testing Convergence of Certain Sums Occasionally I've encountered sums of the form
$$
\lim_{n\to \infty} \sum_{k = 0}^n f(n,k)
$$
Are there known techniques for testing the convergence of such sums? For example, for sums of the form $\sum_{n = 0}^\infty f(n)$ one has the ratio test, the comparison test, the root test, the integral test, and so on. 
 A: You can play around with each of the tests you mention to come up with an extension.
I'll start with one possible extension of the ratio test.
I took some creative liberty in naming the test .
Feel free to ask for hints if you want help extending the others.
Theorem (The Charles Hudgins Ratio Test). Suppose $\limsup_n |a_{n, k}| < \infty$ for each $k$ and $\sup_n \limsup_k |a_{n, k+1} \, / \, a_{n, k}| < 1$. Then, $\lim_n \sum_{k=0}^n a_{n, k}$ converges.
Proof.
By the second assumption, we can pick $K$ large enough so that $|a_{n, k+1}| < r |a_{n, k}|$ for all $k \geq K$ where $0 < r < 1$ is a constant independent of $n$.
For $n$ large enough,
$$
  \sum_{k = 0}^n       \left| a_{n, k} \right|
= \sum_{k = 0}^{K - 1} \left| a_{n, k} \right| 
+ \sum_{k = K}^n       \left| a_{n, k} \right|
< \sum_{k = 0}^{K - 1} \left| a_{n, k} \right|
+ r^{-K} \left| a_{n, K} \right| \sum_{k = K}^n r^k.
$$
The last sum on the right hand side is a convergent geometric series.
As for the first sum, note that
$$
  \limsup_n \sum_{k = 0}^{K - 1}           \left| a_{n, k} \right|
=           \sum_{k = 0}^{K - 1} \limsup_n \left| a_{n, k} \right|
$$
is, by the first assumption, a sum of $K$ finite terms and hence also finite.
