Techniques that let you tell whether a series converges are unimaginatively called tests for convergence or convergence tests. Fortunately for us, we can equate them to Pandora's box. Pandora's box was jam-packed with all of the evils of the world and a little surprise. When she opened it, they all escaped, but the little surprise, hope, remained.
For series, we just have numbers and sandwiches in our box. If the box if full of infinite amounts of food, the series diverges. Grilled cheese goes spills out all over when we open the box. If it has finite amounts, the series converges. The grilled cheese stays put when we open the box.
In this section we'll introduce a lot of convergence tests and show how to use them. We'll also have a great long discussion about how you know when to use which test(s). With our new found knowledge, we will know better than to open up a divergent Pandora's box of grilled cheese.