Key Differences Between Convergent and Divergent Sequences

1. What They Mean:

   • Convergent Sequences: This is when a sequence, which is a list of numbers, gets closer and closer to a specific number called a limit (let's say it's L).

     For example, no matter how small you want the difference between the sequence and the limit (that's what we call epsilon), there will be a point in the sequence after which all numbers are super close to L.

   • Divergent Sequences: This is when a sequence doesn’t get closer to any one specific number as it goes on forever. It can go up, down, or all over the place without settling.

2. Examples of Each:

   • Convergent: Think about the sequence where each term is $a_n = \frac{1}{n}$. As n gets bigger, the terms get closer and closer to $0$. So, we say this sequence converges to $0$.

   • Divergent: Now consider the sequence $b_n = n$. As n keeps growing, this sequence keeps getting larger and larger without stopping. So, we call it divergent because it goes off to infinity.

3. How to Picture Them:

   • Convergent: Imagine a line on a graph that gets closer and closer to a horizontal line (this is the limit).

   • Divergent: Picture a graph that either keeps going higher and higher or swings back and forth without settling down.

In summary, convergent sequences find a limit, while divergent sequences do not!