Sequences and Series
A sequence is a set of numbers in a particular order. For example, {4, 7, 10, 13, 16} is a sequence. It has elements or terms in curly braces (which define it as a set), and the terms are in numerical order (increasing in value). One application of sequences is to add the terms together. We call this a series. Let's review these terms and their definitions:
Sequence: A set of numbers in a particular order.
Series: The sum of the first n terms in a sequence.
To evaluate the series of the sequence {4, 7, 10, 13, 16} we simply add the numbers together. The series evaluates to 50.
Summation Notation
Summation notation is a way to write the sum of sequences. Let's us the sequence {4, 7, 10, 13, 16} to write the series in summation notation:
This notation tells us to sum the values of terms in a sequence, starting with the first term and ending with the fifth term. How was this information drawn out from this notation? The big E-like symbol is the Greek letter sigma. This is what tells us to sum. What exactly are we summing? That's written directly to the right of sigma. a_{n} represents the value of the n-th term to a sequence. Finally, we see expressions above and below sigma. Below is the lower index. This tells us where to start. The 5 above is the upper index, and tells us where to stop summing. So this is telling us to evaluate a_{1} + a_{2} + a_{3} + a_{4} + a_{5}, where each of these terms represents the values of the first, second, third, fourth, and fifth terms of a sequence, respectively.
Summation notation: an expression of a series, using the Greek letter sigma, and a lower and upper index to indicate the first and last terms of the sum.
Evaluating a Series using Summation Notation
Next, let's practice using a defined sequence and summation notation to evaluate a series. Consider our example sequence from before:
Notice, however, that this is now an infinite sequence, meaning that more terms follow 16. How can we interpret what we are asked to sum from the following notation:
We should first note what this is asking us to sum. This is telling us to add values of terms in a sequence. Our sequence is defined as a_{n}, so this is asking us to add together terms in that particular sequence. Next, we should examine the lower index, which tells us what term is the start to our sum, as well as the upper limit, which tell us which term will end the sum. The notation tells us to add together the second through fourth terms of the sequence. This would be 7 + 10 + 13 = 30 as the value of the series.
the sum of the first nth terms in a sequence
a set of numbers in a particular order
an expression of a series, using the Greek letter sigma, and a lower and upper index to indicate the first and last terms of the sum