In University Calculus II, it’s really important to understand whether number sequences come together (converge) or go apart (diverge). This is key to learning more advanced math ideas later on. There are a few common methods to figure out if a sequence converges or diverges. Let’s look at some of these methods.

1. The Limit Test

The limit test is one of the easiest ways to check if a sequence converges. Here’s how it works: look at the limit of the sequence as ( n ) gets really big (infinity).

• If

$\lim_{n \to \infty} a_n = L$

where ( L ) is a regular (finite) number, then the sequence ( (a_n) ) converges to ( L ).

• If

$\lim_{n \to \infty} a_n = \infty$

or does not exist, then the sequence diverges.

For example, the sequence ( a_n = \frac{1}{n} ) converges to 0, while the sequence ( b_n = n ) diverges to infinity.

2. Monotonic Sequence Theorem

A sequence that is either always going up or always going down, and has an upper or lower limit, is guaranteed to converge.

• A sequence ( (a_n) ) is monotonic if it either goes up (( a_{n+1} \geq a_n )) or goes down (( a_{n+1} \leq a_n )) for all ( n ).
• If a monotonic sequence has a limit above or below it, then it converges.

For instance, the sequence given by ( a_n = \frac{n}{n+1} ) goes up and is limited by 1, which means it converges to 1.

3. Squeeze Theorem

The squeeze theorem is handy when you’re not sure how to evaluate some sequences directly. If you can find two sequences ( b_n ) and ( c_n ) such that:

$b_n \leq a_n \leq c_n$

for all ( n ) and

$\lim_{n \to \infty} b_n = \lim_{n \to \infty} c_n = L$

then

$\lim_{n \to \infty} a_n = L$

too.

For example, take ( a_n = \frac{\sin(n)}{n} ). Since we know that ( -1 \leq \sin(n) \leq 1 ), we can say:

$-\frac{1}{n} \leq a_n \leq \frac{1}{n}$

Both ends converge to 0, so ( a_n ) does too.

4. Ratio Test

Even though this is usually used for series, the ratio test can help with some sequences. For a sequence that is defined by a rule, calculate:

$L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|$

• If ( L < 1 ), the sequence converges.
• If ( L > 1 ) or ( L ) is infinite, the sequence diverges.
• If ( L = 1 ), the result isn’t clear.

For example, for the sequence ( a_n = \frac{n!}{n^n} ), using the ratio test shows that it converges.

5. Comparison Test

This method involves comparing your sequence to another sequence that is known to converge or diverge. If ( 0 \leq a_n \leq b_n ), and you know ( (b_n) ) converges, then ( (a_n) ) converges too. On the other hand, if ( (a_n) \geq (b_n) ) and ( (b_n) ) diverges, then ( (a_n) ) also diverges.

For example, if you look at the sequence ( c_n = \frac{1}{n^2} ) (which converges) and compare it to ( a_n = \frac{1}{n^p} ) where ( p > 2 ), you can say that it converges too.

By using these tests, you can better understand how sequences behave. This is an important step in mastering calculus and becoming better at math!