Sigma Notation

Series
Let a₁, a₂, a₃, ⋯, a_n, be a given sequence. Then, the expression

a₁+a₂+a₃+⋯+a_n
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+⋯+a_n is abbreviated as ∑ⁿ_k=1 a_k.

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.

SIGMA NOTATION
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 ∑¹⁰_n=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.

Sigma Notation
u₁+u₂+u₃+u₄+⋯+u_n 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+⋯+u_n as ∑ⁿ_k=1 u_k.
∑ⁿ_k=1 u_k reads “the sum of all numbers of the form u_k where k=1, 2, 3, …, up to n”.

Example 1. Write the sum given by ∑⁷_k=1 (k+5).
solution:

Ex2. Write the sum of the first 25 positive odd numbers in sigma notation.
Solution:
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 a_n=2n-1. The sum of the first 25 odd numbers is ∑²⁵_n=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.
Solution:
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 ∑¹⁰_n=3 (n²+n)
b. Write the series as

3(4)+4(5)+5(6)+6(7)+7(8)+8(9)+9(10)+10(11).
The series is the sum of n(n+1) from n=3 to n=10.
The series can be written as ∑¹⁰_n=3 n(n+1).
Answers: a. ∑¹⁰_n=3 (n²+n) b. ∑¹⁰_n=3 n(n+1)

Ex4. Use sigma notation to write the sum of the reciprocals of the natural numbers.
Solution:
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:

∑^∞_n=1 1/n (Answer)

Series
A finite series is the sum of the terms of a finite sequence. Thus, if

a₁, a₂, …, a_n
is a sequence of n terms, then the corresponding series is
a₁+a₂+⋯+a_n.
The number a_k 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
2+4+6+⋯+100
in which the kth term is given by 2k, then we can write this series as
∑⁵⁰_k=1 2k
Note that the variable k here is a dummy variable. This means that we could also write the series as
∑⁵⁰_i=1 2i or ∑⁵⁰_j=1 2j

Example 5. Write the following sum in sigma notation.

2+4+6+8+⋯+22+24
Notice that we can factor a 2 out of each term to rewrite this sum as
2⋅1+2⋅2+2⋅3+2⋅4+⋯+2⋅11+2⋅12
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 ∑¹²_k=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:

a₁+a₂+a₃+a₄+⋯.
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.

Starting from

S_n=a₁+a₂+⋯+a_n
S_(n-1)=a₁+a₂+⋯+a_(n-1)
by subtracting we obtain
S_n–S_(n-1)=a_n.
For example, if the sum of the first n terms of a series is given by S_n=n², then the nth term is
a_n=S_n–S_(n-1)=n²-(n-1)²=2n-1.
So the terms form the sequence of odd numbers. Hence, we have found a formula for the sum of the first n odd numbers:
1+3+5+⋯+(2n-1)=n².
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.

Sigma Notation
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:

∑ⁿ_i=m a_i =a_m+a_(m+1)+⋯+a_{(n-1)+a_n}
where
● 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;
● a_i 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 S_n– or Σ -notation since they mean the same thing:

S_n=∑ⁿ_i=1 a_i =a₁+a₂+a₃+⋯+a_n
For example, in the following sum,

we have to add together all the terms in the sequence a_i=i from i=1 up until i=5:
∑⁵_i=1 i=1+2+3+4+5=15

Ex6. ∑⁶_i=1 2ⁱ =2¹+2²+2³+2⁴+2⁵+2⁶

=2+4+8+16+32+64=126
Ex7. ∑¹⁰_i=3 (3xⁱ ) =3x³+3x⁴+⋯+3x⁹+3x¹⁰
for any value x.

Ex8. Write out what is meant by

Answer:

Sigma notation
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’.
∑ⁿ_k=1 a_k means ‘the sum of the terms a_k from k=1 to k=n. In other words,

∑ⁿ_k=1 a_k =a₁+a₂+a₃+a₄+⋯+a_n
Example 9.

Properties of the sigma notation
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, a_i and b_i,

2. For any constant c, which is any variable not dependent on the index i,