Sigma Notation of a Series

Wouldn''t it be wonderful if we could represent our series in a more compact, easy-to-read way? After all, writing out terms in a series -- each separated by a + or - sign can become quite tiresome. As it turns out, there is a way to write a series in a more compact form -- and we call this "sigma notation." But what exactly is sigma notation? How do we use it? More importantly, how do we make sense of it? Let''s find out:

What is sigma notation?

The word "sigma" is Greek, and it represents the 18th letter of the Greek alphabet. Our own letter "S" evolved from sigma, and it was first adopted by the Romans. Its symbol is $\sum$ .

In mathematics, it represents "the sum of all terms." When we see the symbol $\sum$ , it means that we''re being asked to "sum up" a sequence. Here''s a useful trick to help us remember what sigma means: The first letter of both "sum" and "sigma" is "s." "S" stands for "sum."

But how exactly do we use sigma notation in mathematics? Whatever comes after the $\sum$ symbol is what we need to sum up. A simple example of sigma notation is as follows:

$\sum$ n This is telling us that we need to find the sum of "n." as n increments by 1 between its starting and stopping values.

Often, we will see symbols on top of the sigma symbol and below. For example:

$\sum _{n=1}^{4}n$

The top number tells us that the series n stops at 4 and begins at 1. In other words, this is asking for the sum of 1, 2, 3, 4.

$1+2+3+4=10$

Hopefully, we are starting to understand that at its core, sigma notation really isn''t so difficult -- even though it involves a strange new symbol!

That being said, sigma notation can be much more complicated -- as we will soon find out.

More complex examples of sigma notation

Consider this example of sigma notation:

$\sum _{n=1}^{6}4n$

Notice that we now have an additional layer of complexity: The coefficient "4" in front of our n.

Most of the basics are still the same. We start with the first value $\left(n=1\right)$ , and we count until the sixth value.

The added coefficient of 4 tells us that the values in our series increase by four each time:

$4,8,12,16,20,24$

We might also express this as successive values of 4 -- but multiplied by successive counting values from 1 to 6:

$4\left(1\right),4\left(2\right),4\left(3\right),4\left(4\right),4\left(5\right),4\left(6\right)=4+8+12+16+20+24$

Either way, the sum of the series is always 84.

But this is only the beginning as we get into sigma notation. It can achieve things with far greater complexity:

$\sum _{n=1}^4{n}^2$

What happens when n is squared? We have to square each successive value in the series:

$1^2+2^2+3^2+4^2$

This leaves us with a total value of 30.

How about this type of sigma notation?

$\sum _{n=1}^4\left(2\mathrm{n}+1\right)$

As we can see, our n value is becoming more and more complex!

$3+5+7+9=24$

Why do we use sigma notation?

Sigma notation can make our lives easier in many different ways. First and foremost, it saves us from having to write out numerous values in a sequence. When we understand the concept of sigma notation, we can write out these series in a matter of seconds.

But in some cases, it might even be impossible to write out all of the values in our series. The obvious example is an infinite series. Even if we had all the time in the world, we couldn''t possibly write out all the values in an infinite series. This is where sigma notation really shines.

We know that we''re dealing with an infinite series when we see a "$\ldots$" at the end of the series -- like so:

$1+2+3+4+5+6+7+\dots$

But how would we write this in sigma notation? Easy. Whenever we see an infinity symbol over the sigma symbol, we know that the series is infinite. Here''s what that might look like:

$\sum _{n=1}^{\infty }n$

Infinite series can be very interesting, and sigma notation helps us conceptualize them. In some cases, we know what the final result must be -- even if we know that the series repeats infinitely. Although this might sound completely impossible, there are a few notable examples, such as:

$\sum _{n=1}^{\infty }{4}^{-n}=\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}+\dots =\frac{1}{3}$

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Sigma notation can be a challenging concept to understand at first, but it can make the rest of your student''s math journey much easier. It makes sense to invest in tutoring for your student, as a math professional can guide them toward confidence in concepts like sigma notation. Tutors can use various teaching methods, and they can choose techniques that match your student''s learning style. They can also set a manageable pace for your student and provide them with plenty of opportunities to ask questions. Reach out to our Educational Directors today for more information, and rest assured: Varsity Tutors will pair your student with a suitable tutor.