In the world of sequences and whether they come together (converge) or drift apart (diverge), there are some important tests that help us understand their behavior. Knowing these tests is really important, especially in a university-level Calculus II class where we look closely at what happens with sequences and series.

Let’s break down some of these tests:

1. Limit Test:
This test is sometimes called the Divergence Test. It tells us that if a sequence, called $(a_n)$, approaches a number $L$, then it’s true that as $n$ gets really big, $a_n$ gets closer and closer to $L$.
But, if $a_n$ doesn’t settle down to a specific number or just keeps getting bigger or smaller without limit, then the sequence diverges. This test helps us get started in figuring out what kind of sequence we have.

2. Monotonicity Test:
This test is great for sequences that move in one direction either up or down.
A sequence is monotonic increasing if each new term is at least as big as the one before it (this means $a_{n+1} \geq a_n$).
A sequence is monotonic decreasing if each new term is at least as small as the one before it (this means $a_{n+1} \leq a_n$).
If a sequence is both monotonic and stays within certain bounds (like not getting too big or too small), then it converges. This idea comes from something called the Monotone Convergence Theorem.

3. Cauchy Criterion:
This test looks at how close terms in a sequence are getting to each other. A sequence $(a_n)$ is Cauchy if, no matter how small of a number you pick (let’s call it $\epsilon$), you can find a point in the sequence (an integer $N$) where all terms after $N$ are really close together.
In simple terms, if the difference between any two terms after a certain point is smaller than $\epsilon$, then it’s Cauchy.
What's cool is that every sequence that converges is also a Cauchy sequence. And in complete spaces (like the real numbers), Cauchy sequences converge.

4. Squeeze Theorem:
This theorem is helpful for sequences that can be "squeezed" between two others that are converging.
If you have two sequences, $b_n$ and $c_n$, such that $b_n \leq a_n \leq c_n$ for all $n$, and both $b_n$ and $c_n$ settle down to the same limit $L$, then $a_n$ will also settle down to $L$.

5. Root and Ratio Tests:
These tests are super helpful when dealing with complicated sequences, especially those that use factorials or exponential terms.
By looking at the ratio or root of terms in the sequence, we can find out if they converge.

In short, these important tests—the Limit Test, Monotonicity Test, Cauchy Criterion, Squeeze Theorem, and Root and Ratio Tests—give us a way to figure out if sequences converge or diverge. They offer students a strong set of tools for solving challenging calculus problems.