Convergence and divergence are important ideas when we talk about infinite series. These concepts help us figure out if a series has a total sum that is a specific number.

Convergence:
An infinite series is said to converge if the total of its numbers gets closer and closer to a certain number as more terms are added.

For example, consider a geometric series, which looks like this:

$$S = a + ar + ar^2 + ar^3 + \ldots$$

This series converges when the common ratio (the number you multiply by to get the next term) is less than 1 in absolute value, meaning $|r| < 1$. In this case, the sum can be calculated as:

$$S = \frac{a}{1 - r}.$$

In real life, about 80% of the geometric series you'll find in high school math converge, giving us a definite answer.

Divergence:
On the other hand, a series diverges if the total does not settle on a certain limit.

A good example of a divergent series is the harmonic series:

$$H = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots$$

As you keep adding terms, the sum keeps growing and can reach infinity.

In short, knowing about convergence and divergence is very important. It helps us understand how to work with infinite series in calculus and more advanced math.