Reasoning: An Introduction to Logic, Sets, and Functions

Subsection 8.8.1 Summation Notation

Suppose that we want to talk about the sum of some sequence of values. One way to do that is to write something of the form:

v1 + v2 + v3 + v4 + …

For example, we could describe the sum of the first 100 positive integers as: 1 + 2 + 3 + 4 + 5 + … + 100

But what if it is not completely obvious to all of our readers exactly what the rest of the … sequence is? Using this notation can be risky.

So we need a clearer, more formal notation. The one that we’ll use exploits Σ (Sigma). To use Σ, we must specify:

• A name for a placeholder variable that we can use to describe values,

• The starting value for that variable,

• The ending value for that variable, and

• A description of the values to be added.

We specify these things by writing a statement of this form:

We’ll read such an expression as, “The sum, as <variable> goes from <starting value> to <ending value> of <expression>. Of course, when we actually do this, we don’t write or say “variable” or “starting value” or “ending value” or “expression”. We substitute specific values. Note that the starting and ending values themselves are explicitly included.

For example, we can describe the sum of the first 100 positive integers as:

\begin{equation*} \sum_{100 }^{i=1}{i} \end{equation*}

The expressions that we write for both starting value and ending value can be written in a general form, in terms of other variables whose values we plan to fill in later.

For example, we can describe the sum of the first 100 positive integers as:

\begin{equation*} \sum_{n }^{i=1}{i} \end{equation*}

The sum is performed on the values in the range, starting value … ending value. But we can add constraints so that not all values in that range are included.

For example, here’s a description of the sum of just the odd integers, starting at 1 and going up to 10:

\begin{equation*} \sum_{10 }^{i=1}{i} \end{equation*}

In other words, 1 + 3 + 5 + 7 + 9 = 25.

The expression that we sum can be arbitrarily complex.

For example, here’s a description of the sum of the first n values that are one less than some integer power of 2:

\begin{equation*} \sum_{n}^{i=1}{2^i - 1 } \end{equation*}

In other words (21 – 1) + (22 – 1) + (23 -1) + (24 – 1) + … (2n – 1).