When we learn about sequences in Grade 10 Pre-Calculus, we want to find out if they converge or diverge.

• Converging means that they get closer to a specific number.
• Diverging means they don't settle around any number.

There are several easy tests we can use to see if a sequence converges.

1. The Limit Test

One of the simplest ways is to look at the limit of the sequence as ( n ) gets really big.

If we find that: $\lim_{n \to \infty} a_n = L$

and ( L ) is a regular number (not infinity), then the sequence converges to ( L ).

For example, take the sequence defined by:

$a_n = \frac{1}{n}.$

As ( n ) becomes really large:

$\lim_{n \to \infty} \frac{1}{n} = 0,$

so this sequence converges to 0.

2. Monotonicity Test

If a sequence is monotonic, which means it either always goes up or always goes down, and it is bounded (stays within a certain range), then it will converge.

For instance, the sequence:

$a_n = \frac{n}{n+1}$

always increases and gets closer to 1 as ( n ) gets larger. So, it converges to 1.

3. The Squeeze Theorem

This theorem helps us find limits by placing a sequence between two other sequences that we already know converge to the same limit.

If:

$b_n \leq a_n \leq c_n$

and both ( b_n ) and ( c_n ) get closer to ( L ), then according to the Squeeze Theorem, ( a_n ) also converges to ( L ).

By using these tests, we can easily tell whether sequences converge or not. This helps us understand sequences and series better!