In this explainer, we will learn how to express a series in sigma notation and how to expand and evaluate series represented in sigma notation.
In mathematics, a sequence can be loosely thought of as an ordered list of numbers. Given this very open-ended and vague description, it should be easy to believe that there are infinitely many sequences. Often sequences of interest will relate to basic concepts within mathematics that can be described using this idea. For example, a common sequence of numbers would be the square numbers, whereby every positive integer is squared and presented in order, giving 1, 4, 9, 16, 25, etc. Given that there are infinitely many square numbers, the sequence of these is infinite. However, we could choose a finite subset of these numbers such as 16, 25, 36, 49, and this would still constitute a sequence based on the square numbers, albeit a finite sequence.
Once a sequence has been well defined, it can be used to create a “series”, which essentially consists of adding together the elements of a sequence in the original order. For example, if we were given the sequence above as 16, 25, 36, 49, then the series corresponding to this sequence would be , and, in this case, the series can be evaluated to give a total of 126. Evaluating a series can prove challenging and it may be impossible to do this using a closed-form expression, depending largely on the form of the sequence that the series is based on.
Sigma notation is a convenient way of representing series where each term of the summation can be defined by a sequence or function. There are many important types of series that appear across mathematics, with some of the most common being arithmetic series and geometric series, both of which can be represented succinctly using sigma notation. Sigma notation can be used to encode any map or function where the output is derived from adding together the terms of a given sequence and is rich with algebraic properties that can be used to simplify calculations through this highly condensed notation.
Definition: Sigma Notation
Consider a function or sequence that is denoted , where . Then, for two integers and , where , we can sum over this function using the sigma notation
This expression dictates that we evaluate for and then add together these values in order. Assuming that and are finite, the expanded form of this expression can be written as
Often and are respectively referred to as the “lower limit” and the “upper limit” of the series. It does not have to be the case that and are finite, although normally at least one of these expressions will be. However, if at least one of these two limits is infinite then we will be adding together an infinite numbers of terms and, understandably, this situation can be a lot more challenging than when there are only a finite number of terms to evaluate. Most typically, series using sigma notation will involve being equal to 1. However, it is entirely possible that the lower limit is not equal to 1, and there are situations where it becomes convenient to alter the lower and upper limits to improve the brevity and neatness of subsequent calculations.
We will illustrate this concept with a simple example. Suppose that we undertook to evaluate the following expression written in sigma notation:
First, we should note that the lower limit is 1 and the upper limit is 5. The sequence that is used to generate this series can be written as . By now evaluating this function for every integer value between (and strictly including) the two limits, we would obtain the following:
The complete series can hence be expressed by adding together all of these terms in order, thus giving
In reality, there is seldom a need to write out the series in terms of the function , and usually we would choose to omit the first line of working in the equations immediately above. However, this definition is useful when looking to complete the process in reverse, that is, by inspecting a series written out in long form and attempting to write this in concise sigma notation. We will see several examples of this later in the explainer, after practicing a few instances of the above technique.
Example 1: Evaluating the Sum of a Finite Series after Expanding It
Expand and then evaluate .
First, we choose to express the sequence as , noting that the lower limit is 1 and the upper limit is 4. It will benefit us to write out the value of this sequence for all integer inputs between (and including) the lower limit and the upper limit. Doing this gives
With these terms written down, it is now a simple matter to express the full series as
The above example is a fairly simplistic extension of the very first example that we gave in this explainer, whereby the generating sequence contained only one term. This example was deliberately chosen so as to demonstrate the concept involved, rather than require much in the way of lengthy or tedious calculations. It is possible, of course, to use much more elaborate sequences to generate the series using sigma notation. Providing that the same method is applied, there is little reason as to why this should be conceptually more difficult, although of course the calculations involved are likely to take longer and be harder to complete accurately. In the following question we will give a slightly more complicated example, where the sequence used has two terms.
Example 2: Evaluating the Sum of a Finite Series after Expanding It
Expand and then evaluate .
We begin by classifying the expression above in terms of the sequence . The lower limit of the variable is 1 and the upper limit is 4. This means that we should write out the full expressions
The full series can then be written as
In the previous two examples, we have had a lower limit of 1. Although a lower limit of either 0 or 1 is the most common, both this limit and the upper limit may take any integer value (providing that the lower limit is less than or equal to the upper limit). Our third example is similar to the two examples above in that the series is fairly simple, although the lower limit is not equal to 1. As we will see, if we apply the same method as in the previous examples, then this type of question is no longer difficult to answer.
Example 3: Evaluating the Sum of a Finite Series
To assist with evaluating the above series, we define the sequence and observe that the lowest limit of the variable is 3 and the upper limit is 5. We then compute the following:
With these individual terms now calculated, we can evaluate the summation as follows:
So far in this explainer, we have dealt with several examples where we are given an expression in sigma notation and have been asked to evaluate it. We achieved this by writing out each term of the series and adding them together in order. Assuming that we are proficient with sigma notation, functional representation, the ancient skill of addition, and so on, this task is usually of limited difficulty. In reality, we would often choose to evaluate these expressions using computers or other specific calculators, especially if the series contains many terms or is particularly complicated.
The reverse process is when we are presented with the summation written out in full and asked to represent this using concise sigma notation. Generally speaking, this task is more difficult to complete and is more prone to errors being made. That being said, there are several rules of thumb that can be applied to ensure that this task is as painless as possible. Generally speaking, we will always aim to write a series in sigma notation after having first factored the expression as much as possible, thereby reducing the complexity involved. If this process is completed thoroughly, then the expression in terms of sigma notation is normally fairly quick to find. We will begin with a very simple example where no factoring is needed.
Example 4: Expressing a Given Series in Sigma Notation
Express the series in sigma notation, .
It is likely that we are interested in the sequence , where the lower limit is 26. This would suggest that the series should be written in sigma notation as
We might see immediately that this is the correct expression. However, to fully demonstrate that this is correct, we define the function and then complete the following working: which is indeed the original series.
Example 5: Expressing a Given Series in Sigma Notation
Express the series in sigma notation.
In the series above we note immediately that each terms contains a factor of 54, allowing us to instead write
In this new form, we can see that we can actually simplify the right-hand side even further by removing the factor of 12 that appears in every term, thus giving
This shows that the series can be written in terms of the sequence with a lower limit of 1 and an upper limit of 20. Therefore,
In the previous example, we saw how an apparently complicated series can be reduced to a simple expression involving the sequence , much like the example that preceded this. It is often the case that series will involve common sequences such as , or others such as , where is an integer. Even when this is the case, there is still generally a benefit of attempting to deduce any factors that are common to every term of a series, which will show whether a simpler form will be admitted. We give an example of this in the following question.
Example 6: Expressing a Given Series in Sigma Notation
Express the series in sigma notation.
Upon first inspection, we can see that every term of the sequence appears to involve a factor of 8. We assume that this is the case and complete the factorization as follows:
The series within the brackets is now very familiar, since these numbers are the square numbers with a lower limit of 1 and an upper limit of 8. This observation gives
Thus the series is generated by the sequence with a lower limit of 1 and an upper limit of 8, meaning that the full expression in terms of sigma notation is
Occasionally, when working with finite series, there will be multiple ways of representing them using sigma notation. Often, these ways involve making some alterations to the lower and upper limits or factoring in creative ways. This process is aided if we are given the th term of the series in algebraic form. As we will see in the following example, there can be multiple ways to express a given summation, depending on our preference.
Example 7: Expressing a Given Series in Sigma Notation
Express the series in sigma notation.
The term suggests that we might be seeking a sequence of the form . We can immediately see that this was a very good guess, and by evaluating this sequence for the first few positive integers, we find
These give exactly the first three terms in the original series, meaning that we can write this in terms of sigma notation as
Given that we defined , the series can be written in sigma notation as
In the previous question we easily could have chosen to express the series in a slightly different format. For example, we could choose to express the function in the simplified format , which would have given the expression
As an exercise for the reader, we could also modify the lower and upper limits of the summation to give the equivalent expression
We will not demonstrate how exactly we achieved this expression, but the equivalence with the original series can be checked by expanding out the summation on a term-by-term basis.
So far in this explainer, we have only worked with series that contain a finite number of terms. In the following example, we will work with an upper limit that is nonfinite. In this explainer, we are not concerned with evaluating such series (In this case, there is no possible evaluation, as the series diverges), but instead we will only be focusing on writing the series in sigma notation. We will need to borrow inspiration from the previous examples and, as we will see, there are at least two ways in which we might choose to express the series in terms of sigma notation.
Example 8: Expressing a Given Series in Sigma Notation
Express the series in sigma notation.
For the sake of argument, we will go through this example with the aim of writing this series in sigma notation, where the lower limit is equal to 1. This is ultimately an arbitrary choice, but it is the most conventional value to pick in the absence of other information. We are therefore looking to create a sequence that generates each term of the series if the lower limit is 1. In other words, we will try to find such that
Writing out each term in the series in this way, it becomes, by observation, clear that the sequence should be of the form , and we can check this for every given sequence value. With this understood, we can then write the series as
where the upper limit of represents the fact that the series is not truncated after a finite number of terms. Given that we defined , we then have
As with the previous example, the series in the example immediately above can be represented in multiple ways. By altering the lower and upper limits of the summation, we could equally have chosen to represent this series using sigma notation as follows:
Alternatively, if we were willing to be more adventurous, then we could have chosen the equivalent expression
The particular choice of expression depends very much on individual preference, with some people preferring to have a neater sequence by being willing to alter the lower or upper limits away from the conventional choices. When choosing an alternative expression where the limits are manipulated, then it is always sensible to expand the series to ensure that this matches term-for-term with any equivalent expression.
When writing a series in sigma notation, it can be helpful to see how each term can be related to common sequences such as power series, exponential series, and so on. For questions that appear in examinations and textbooks, a sequence can look complicated and messy while actually being fairly simplistic when considered in relation to simpler, known sequences. The following example will demonstrate this concept and how a little flexibility can help to understand sequences that otherwise appear to be complicated or intractable.
Example 9: Expressing a Given Series in Sigma Notation
Express the series in sigma notation to terms.
We begin by assuming that there is a sequence that generates the above series, which we will define as . Furthermore, in the absence of other information, we will assume that the lower limit of the summation is 1 and that the upper limit is . Although we do not yet have a way of writing , we can see by examining each term that it is related to the powers of 10 as follows:
Written out in this way, it is clear that and hence . We can evaluate , and to check that this does indeed reproduce the first four terms of the series. This allows us to see that evaluating the original series to terms gives
We have now covered a range of examples in this explainer that should provide an initial grounding in sigma notation. However, we have largely neglected to discuss the algebraic rules governing sigma notation, many of which would have allowed the examples above to be completed with greater speed and accuracy. Learning to efficiently manipulate expressions in sigma notation is a critical part of understanding mathematics at a higher level, and a fluency with this topic will reap rewards in other areas such as calculus and linear algebra. This being the case, it is still possible to move in and out of sigma notation without understanding these higher-level techniques, providing that this is done methodically and that every term of the sequence is checked for accuracy. As shown in several of the above examples, there are usually multiple ways of expressing a summation in sigma notation by altering the limits or the algebraic form of the generating sequence.
- Sigma notation encodes the following series: where is the lower limit and is the upper limit.
- It must be the case that , although there is no requirement that or is finite.
- In the absence of other information, we would normally assume that the lower limit is 1.
- Although it is not necessary, it is often helpful to calculate the values of before beginning to evaluate a series written in sigma notation.
- When adding together the separate terms of a series, this must be completed in order. Strictly speaking, this is not necessary when and are both finite, but it is a matter of good practice.
- Trying to write a full series in sigma notation can be difficult, but it is greatly aided by factoring the full summation as much as possible.
- There are usually multiple sensible ways of expressing a summation in sigma notation.