Let a1, a2, a3, ⋯, an, be a given sequence. Then, the expression
is called the series associated with the given sequence. The series is finite or infinite according as the given sequence is finite or infinite. Series are often represented in compact form, called sigma notation, using the Greek letter Σ (sigma) as means of indicating the summation involved. Thus, the series a1+a2+a3+⋯+an is abbreviated as ∑nk=1 ak.
Remark: When the series is used, it refers to the indicated sum not to the sum itself. For example, 1+3+5+7 is a finite series with four terms.
When we use the phrase “sum of a series”, we will mean the number that results from adding the terms, the sum of the series is 16.
Writing out a series can be time-consuming and lengthy. For convenience, there is a more concise notatiogi called sigma notation. The series 3+6+9+12+⋯+30 can be expressed as ∑10n=1 3n. This expression is read the sum of 3n as n goes from 1 to 10.
The variable, in this case n, is called the index of summation.
To generate the terms of a series given in sigma notation, successively replace the index of summation with consecutive integers between the first and last values of the index, inclusive. For the series above, the values of n are 1, 2, 3, and so on, through 10.
u1+u2+u3+u4+⋯+un can be written more compactly using sigma notation. The symbol Σ is called sigma. It is the equivalent of capital S in the Greek alphabet.
We write u1+u2+u3+u4+⋯+un as ∑nk=1 uk.
∑nk=1 uk reads “the sum of all numbers of the form uk where k=1, 2, 3, …, up to n”.
Example 1. Write the sum given by ∑7k=1 (k+5).
Ex2. Write the sum of the first 25 positive odd numbers in sigma notation.
The positive odd numbers are 1, 3, 5, 7, ⋯.
The 1st positive odd number is 1 less than twice 1, the 2nd positive odd number is 1 less than twice 2, the 3rd positive odd number is 1 less than twice 3. In general, the nth positive odd number is 1 less than twice n or an=2n-1. The sum of the first 25 odd numbers is ∑25n=1 (2n-1). (Answer)
Ex3. Use sigma notation to write the series 12+20+30+42+56+72+90+110 in two different ways:
a. Express each term as a sum of two numbers, one of which is a square.
b. Express each term as a product of two numbers.
a. The terms of this series can be written as 32+3, 42+4, 52+5, ⋯, 102+10, or, in general, as n2+n with n from 3 to 10.
The series can be written as ∑10n=3 (n2+n)
b. Write the series as
The series is the sum of n(n+1) from n=3 to n=10.
The series can be written as ∑10n=3 n(n+1).
Answers: a. ∑10n=3 (n2+n) b. ∑10n=3 n(n+1)
Ex4. Use sigma notation to write the sum of the reciprocals of the natural numbers.
The reciprocals of the natural numbers are 1, ½, ⅓, ¼, ⋯, 1/n.
Since there is no largest natural number, this sequence has no last term. Therefore, the sum of the terms of this sequence is an infinite series. In sigma notation, the sum of the reciprocals of the natural numbers is:
A finite series is the sum of the terms of a finite sequence. Thus, if
is a sequence of n terms, then the corresponding series is
The number ak is referred to as the kth term of the series.
We often use the sigma notation for series. For example, if we have the series
in which the kth term is given by 2k, then we can write this series as
Note that the variable k here is a dummy variable. This means that we could also write the series as
Example 5. Write the following sum in sigma notation.
Notice that we can factor a 2 out of each term to rewrite this sum as
That means that we are adding together 2 times every number between 1 and 12. The sigma notation could be
There is no need to use k as our index variable. We could have just as easily used m or j instead.
Notice, that these are NOT the same as ∑12k=1 2m
Study Tip: Sigma Notation
There are many ways to represent a given series.
An infinite series is the ‘formal sum’ of the terms of an infinite sequence:
For example, the sequence of odd numbers gives the infinite series 1+3+5+7+⋯.
We can sum an infinite series to a finite number of terms. The sum of the first n terms of an infinite series is often written as
This is sometimes called the nth partial sum of the infinite series.
Given a formula for the sum of the first n terms of a series, we can recover a formula for the nth term by a simple subtraction, as follows.
by subtracting we obtain
For example, if the sum of the first n terms of a series is given by Sn=n2, then the nth term is
So the terms form the sequence of odd numbers. Hence, we have found a formula for the sum of the first n odd numbers:
In general, it can be difficult to find a simple formula for the sum of a series to n terms.
For the rest of this section, we restrict our attention to arithmetic and geometric series.
In this section we introduce a notation that will make our lives a little easier.
A sum may be written out using the summation symbol Σ. This symbol is sigma, which is the capital letter “S” in the Greek alphabet. It indicates that you must sum the expression to the right of it:
● i is the index of the sum;
● m is the lower bound (or start index), shown below the summation symbol;
● n is the upper bound (or end index), shown above the summation symbol;
● ai are the terms of a sequence.
The index i is increased from m to n in steps of 1.
If we are summing from n=1 (which implies summing from the first term in a sequence), then we can use either Sn– or Σ -notation since they mean the same thing:
For example, in the following sum,
we have to add together all the terms in the sequence ai=i from i=1 up until i=5:
Ex6. ∑6i=1 2i =21+22+23+24+25+26
Ex7. ∑10i=3 (3xi ) =3x3+3x4+⋯+3x9+3x10
for any value x.
Ex8. Write out what is meant by
Here is another useful way of representing a series.
The sum of a series can be written in sigma notation. The symbol sigma is a Greek letter that stands for ‘the sum of’.
To determine the number of terms: top value mihus bottom value plus 1 i.e the number of terms in this case is (17-3)+1+15.
Σ is the symbol for ‘the sum of’.
∑nk=1 ak means ‘the sum of the terms ak from k=1 to k=n. In other words,
There are a number of useful results that we can obtain when we use sigma notation.
1) For example, suppose we had a sum of constant terms
What does this mean? If we write this out in full, we get
In general, if we sum a constant 11 times then we can write
2) Suppose we have the sum of a constant times i. VVhat does this give us? For example,
However, this can also be interpreted as follows
which implies that
In general, we can say
3) Suppose that we need to consider the summation of two different functions, such as
Some Basic Rules for Sigma Notation
1. Given two sequences, ai and bi,
2. For any constant c, which is any variable not dependent on the index i,
There are a number of useful results that we can obtain when we use sigma notation. For example, suppose we had a sum of constant terms
What does this mean? If we write this out in full then We get
In general, if we sum a constant n times then we can write
Suppose we have the sum of a constant times k. What does this give us? For example,
But we can see from this calculation that the result also equals
In general, we can say that
Suppose we have the sum of k plus a Constant. What does this give us? For example, 4
But we can see from this calculation that the result also equals
In general, We can say that
Notice that we have written the answer with the constant no on the left, rather than as
to make it clear that the sigma refers just to the k and not to the constant nc. Another way of making this clear would be to write
In fact we can generalise this result even further. If we have any function g(k) of k, then we can write
by using the same type of argument, and we can also write
where a is another constant. We can also consider the sum of two different functions, such as
In general, we can write
and in fact we could even extend this to the sum of several functions of k.
Key Point: If a and c are constants, and if f(k) and g(k) are functions of k, then
We shall finish by taking a particular example and using sigma notation. Suppose that We want to find the mean of a set of examination marks. Now
So if the marks were 2, 3, 4, 5 and 6, we would have
But more generally, if we have a set of marks xm, where i runs from 1 to n, we can write the mean using sigma notation. We
Ex10. By writing out the terms explicitly, show that