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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 + ...
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.
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.
Since we've never made an infinitely large brownie before, we need to figure it out. Hopefully we can find enough flour for this tantalizing undertaking. We're going to figure out how to make an infinite sum the same way.
We can write down all the terms of the series that sums the numbers from 1 to 10:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10.
This series is short, but writing down all the terms of a series is only reasonable when we don't have many terms.
If we want to add the numbers from 1 to 10,000, writing all the numbers down is ridiculous.
There are two reasonable ways we could write the sum.
We could use dots to mean "keep going" like this:
1 + 2 +...+ 9999 + 10000.
We'll call this the expanded form of the sum.
We could use sigma notation, also known as summation notation.
Sigma is a Greek letter. An uppercase sigma looks sort of like a large pointy "E":
A lowercase sigma looks sort of like a lowercase "o" with a tail:
We use uppercase Σ to write down both finite and infinite sums in a compact and tidy way. When Σ is being used this way we call it the summation sign.
Taking the sum
1 + 2 +...+ 9999 + 10,000
as an example. We're going to write this sum using sigma notation.
Since we want to add a bunch of things together, we start by writing the summation sign
This means add things. The summation sign is like our mixer and mixing bowl for our brownies.
What are we adding? We're adding a bunch of terms. The 1st term is a1 = 1, the 2nd term is a2 = 2, and the general nth term is an = n.
We write the general term inside the summation sign:
This expression means 'add up the values n.' Here, each term is another ingredient to our brownie recipe.
We need a little more context. We can't make brownies if we don't know which and how many ingredients we need. Which values of n are we supposed to add up? We want to start at n = 1, so we write that at the bottom of the summation sign:
We want to stop when n = 10,000, so we write that at the top of the summation sign:
Now we have the expression
which means "the sum of n from 1 to 10000." This expression means the same thing as the sum we started with:
If this were a brownie recipe, we'd need 10,000 different ingredients.
We can see that we need three things to write a series using sigma notation:
1) the general term of the series, an,
2) the starting value of n, and
3) the ending value of n (if the series is finite).
We find the general term of the series, an, the same way we find the general term of a sequence.
The starting and ending values of n, written at the top and bottom of the summation sign, are called the limits of summation. These numbers tell us which terms we're adding up. The variable n is called the index of summation, since it indexes which term we're on.
If we're baking a finite brownie, we have a finite list of ingredients. Likewise, a finite series written in sigma notation will have finite limits:
Write the sum
4 + 9 + 16 + ... + 1000
using summation notation.
We're adding squares:
22 + 33 + ... + 102
The general term looks like n2. We want to add up the terms n2, so we write
We start with the term 22 and end with the term 102, so we're going from n = 2 to n = 10. We write these as the limits of summation:
We want to bake an infinite brownie, so we need an infinite list of ingredients. Our infinite series will have ∞ instead of a final value for n. This shows that we're adding infinitely many terms. In sigma notation, we write the starting value below the summation sign as usual, but write ∞ instead of an ending value.
With sequences, sometimes we like to start at n = 0 and sometimes we like to start at n = 1. Similarly, we can use sigma notation starting at different values of n to write the same series in different ways. Sometimes we start at n = 0, sometimes at n = 1, and sometimes at other numbers. With our infinite brownie, it doesn't matter where it begins because we will have an infinitely large tasty treat to indulge ourselves.
When we use different limits, we need to be careful to define our general term properly. We can change the beginning limit from n = 1 to n = 6, but if we don't change the general term, our series will not be the same. That'd be like using sand instead of flour in our brownie recipe. Not quite as tasty.