When you start learning about sequences and series, you find some really interesting examples of convergent and divergent series. These examples help you understand what convergence means.

Famous Convergent Series:

1. Geometric Series: A series like $\sum_{n=0}^{\infty} ar^n$ converges, which means it adds up to a specific number, if the absolute value of $r$ is less than 1.

   For example, $\sum_{n=0}^{\infty} \left(\frac{1}{2}\right)^n$ adds up to 2.

2. P-Series: This series is written as $\sum_{n=1}^{\infty} \frac{1}{n^p}$. It converges if $p$ is greater than 1.

   A well-known example is $\sum_{n=1}^{\infty} \frac{1}{n^2}$, which converges to a special number known as $\frac{\pi^2}{6}$.

Famous Divergent Series:

1. Harmonic Series: The series $\sum_{n=1}^{\infty} \frac{1}{n}$ diverges. This means that as you keep adding more terms, the total keeps growing without stopping.

2. Alternating Harmonic Series: Interestingly, the series $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$ converges to a number called $\ln(2)$, even though it alternates between positive and negative values.

These examples beautifully show the ideas of convergence and divergence in math!