When exploring the world of infinite series, I've discovered some important features that help us tell them apart. Here are a few things that make them special:

1. Convergence vs. Divergence:

   • One major difference is whether an infinite series converges or diverges.
   • If a series converges, it means it adds up to a specific number. An example is the series $\sum_{n=1}^{\infty} \frac{1}{n^2}$, which adds up to something finite.
   • On the other hand, a series diverges when it keeps growing forever. For instance, the series $\sum_{n=1}^{\infty} n$ diverges.

2. Types of Series:

   • There are different kinds of series out there.
   • A geometric series is one where each term is a constant times the previous term. The common formula is $\sum_{n=0}^{\infty} ar^n$. It converges if $|r| < 1$.
   • Another type is the harmonic series, which is written as $\sum_{n=1}^{\infty} \frac{1}{n}$. This one diverges!

3. Alternating Series:

   • These series switch back and forth in sign, like $\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n}$.
   • They can converge if certain conditions are met, which we can check using the Alternating Series Test.

4. Rate of Convergence:

   • Lastly, there’s the speed of convergence, which is how fast a series gets close to its limit.
   • Some series, like the one for $e^x$, converge quickly, while others might take longer.

Understanding these features can help you grasp the idea of infinite series and see how they behave!