Series: This is the Sum That Doesn't End

If you've never heard "The Song that Doesn't End", don't follow this link, we dare you.

Never-ending series are much less annoying and much more useful. We promise.

A series is a sum of numbers such as

1 + 2 + 3 + 4 + ...

or

0 + (-1) + (-2) + (-3) + (-4) + ...

The individual numbers are called the terms of the series.

Each term can be positive, negative, or zero.

Be Careful: In math, a sequence and a series are not the same thing.

In a sequence the terms are separated by commas:

1, 2, 3, 4,....

In a series the terms are separated by plus signs:

1 + 2 + 3 + 4 + ...

A series is what we get when we add all the terms of a sequence. We'll talk more later about what it means to add infinitely many numbers.

So we sort of lied about series being an infinite sum, but we didn't Houdini you. Series can be finite or infinite, but we can turn any finite series into an infinite series by attaching infinitely many zeroes to the end.

Sample Problem

The finite series

1 + 2 + 3

is the same as the infinite series

1 + 2 + 3 + 0 + 0 + 0 + 0 + ...

Language has two components: syntax and semantics. Syntax is how we write things. It is our words, sentence structure and punctuation. Although it may not have been our favorite subject in school, without it we wouldn't be able to enjoy the semantics.

Semantics are what we mean when we say or write something. When you walk up to a stranger and say, "It's a beautiful day outside. Isn't it?" you are also starting a conversation to say hello.

Math also has syntax and semantics. To talk about series we need to address both. Let's stick one to Zeno in the process.

• Syntax: How do we write down an infinite sum? Put another way, how do we draw a brownie that is infinite is size?

• Semantics: What does it mean to add infinitely many numbers? Or, what would an infinitely large brownie look and taste like?