Video Lecture

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Quick, write down the expression to add up all the numbers between one and a thousand. How long do you think would that take? Probably a decent chunk of time and mathematicians are, if nothing else, way too lazy to do something so repetitive. Nonetheless, writing out gigantic sums like this is actually something that is not only useful, but will soon be necessary for our work in calculus. So, it is time we introduced a new notation that mathematicians use to represent repeated addition like this... enter sigma notation.

Sigma notation is named after the capital Greek letter that is used, $\Sigma$. It’s worth a mention that there is, technically speaking, no reason we need to wait until calculus to introduce this notation - it does not inherently involve any machinery of calculus. Rather, we just haven’t really needed it until now, which is why we haven’t introduced it yet.

As mentioned, Sigma Notation is a way to write repeated sums, but an important aspect of this is that there needs to be a formula that tells you which numbers need to be added together. Let’s start with a look at the notational form, and then see some examples of how it works.

Sigma notation has the following form:

$$\sum \limits _{\text {Index}=\text {Start}}^{\text {End}} \text {Formula of Index}$$

Sigma notation uses the capital sigma letter, with an index below it, a value above it, and a formula to the right which represent an algorithm to determine what values are added together. This is a bit easier to understand with a concrete example. Consider the following;

$$\sum \limits _{k=1}^{1000} k$$

In this case, our index is the letter $k$, which starts at $1$ and continues to $1000$. The formula is simply “$k$”. This would be read as “the sum from $k$ equals $1$ to $1000$ of $k$.”

So, how does the notation actually work? Simply put, one plugs in the starting value of the index everywhere we see the index letter in the formula, then once you compute that result, you increase the index by 1 and then do it again, adding it to the previous value. You iterate this process by increasing the index value by 1 each time, until you reach the “End” value, which is the last index value you plug in, calculate, and add to the running sum.

So, in our example of $\displaystyle \sum \limits _{k=1}^{1000} k$, we would start by plugging in $1$ for “$k$” (the index) in the formula. Since the formula is only $k$, this results in $1$. Then we increase $k$ by 1 and do it again, adding the previous value... so we increase $k$ to $2$, then plug it into the formula (again, the formula is just $k$, so) we get “$2$”, then we add that to the previous value of $1$. Then we increase $k$ yet again to $3$ and plug it into the formula to get $3$, and add it to the previous value, giving us $1 + 2 + 3$ and so on and so on... For this sum we would get $1 + 2 + 3 + \dots + 999 + 1000$. Once we reach $1000$ we have hit the “End” value and so we plug that value into the formula as the last step, adding it to the running sum we’ve been writing and stopping there.

So, we can write the original goal (the sum from $1$ to $1000$) as $\displaystyle \sum \limits _{k=1}^{1000} k$.

Some important notes here:

• The index letter doesn’t particularly matter, but by convention they tend to be one of $i, j, k, l, n$ or $m$. They are typically lowercase, but importantly they should not overlap any other letter in the formula (like variables) if the formula contains any letters.
• The index value must always be an integer. It’s possible to have negatives or positives (we’ll see some examples shortly) but there cannot be fractional, or irrational, index values - they must always be integer values at the start and end values.
• The formula is typically some function of the index letter, although it is possible to have formulas that don’t involve the index at all. Nonetheless, the index still needs to be defined as part of the notation.
• The sigma notation represents a sum, which means that you can do with it what you can do with most sums unless the sum is infinite. Once infinity makes an appearance, all intuition and rules generally no longer apply. We won’t be dealing with this situation too often (although it will come up in this class), but this is an entire area of study in calculus two, so it is important to at least make a note of it here.

Ok, lets see a few examples to get a good handle on how the notation works. We will write the notation version, and then the “long hand” version of several sums to show how it works.

• $\displaystyle \sum \limits _{k=0}^5 2k = 2(0) + 2(1) + 2(2) + 2(3) + 2(4) + 2(5) = 2 + 4 + 6 + 8 + 10 = 30$.
• $\displaystyle \sum \limits _{n=-3}^{-1} n^2 + 1 = \bigg ((-3)^2 + 1\bigg ) + \bigg ((-2)^2 + 1\bigg ) + \bigg ((-1)^2 + 1\bigg ) = (10) + (5) + (2) = 17$.
• $\displaystyle \sum \limits _{i=3}^9 2 = 2 + 2 + 2 + 2 + 2 + 2 + 2 = 14$.
• $\displaystyle \sum \limits _{m=-1}^3 2m^2 + m - 1 = \bigg (2(-1)^2 + (-1) - 1\bigg ) + \bigg (2(0)^2 + (0) - 1\bigg ) + \bigg (2(1)^2 + (1) - 1\bigg ) + \bigg (2(2)^2 + (2) - 1\bigg ) + \bigg (2(3)^2 + (3) - 1\bigg ) = (0) + (-1) + (2) + (9) = 10$.
• $\displaystyle \sum \limits _{j=11}^{13} 2^{j-10}+1 = \bigg ( 2^{(11) - 10} + 1 \bigg ) + \bigg ( 2^{(12) - 10} + 1 \bigg ) + \bigg ( 2^{(13) - 10} + 1 \bigg ) = 3 + 5 + 9 = 17$.

We have introduced the sigma notation which, although technically not a calculus topic directly, is something we will soon need as we continue in our exploration of calculus. This notation allows us a nice condensed way of writing long sums whose terms can be algorithmically described with some kind of formula, along with an iterating index. We’ll see lots more examples of this notation (and why it can be so helpful) as we move forward into Riemann sums, but for now we aimed to introduce the basics of the notation and how it works mechanically.