College Algebra
Tutorial 54B: Series
 Learning Objectives
After completing this tutorial, you should be able to:
- Know how to evaluate a summation.
- Find the sum of a series.
- Write a series in summation notation.
- Find the mean of a sequence of numbers.
|
Introduction
In this tutorial we will mainly be going over
series. We will start by going through some basic terminology
associated with series. In a series you are working with the sum
of terms of a sequence. If you need a review on sequences, feel
free to go to Tutorial
54A: Sequences. Arithmetic and geometric series are
special forms that are looked at more
in depth in Tutorial 54C: Arithmetic Sequences and Series and Tutorial 54D: Geometric Sequences and Series.
We will be looking at
series forwards and backwards. Once you are able to go back and
forth,
then that means you have series down. Enough of that, let's
get started. |
Tutorial
Summation Notation
|
The greek
letter sigma is used to indicate a summation.
Summation notation is a shorthand way of saying take the sum of certain
terms of a sequence.
Let’s look at the following:
In this example, i represents
the term number or index of summation. Note that any variable can
be used here.
 represents the general term.
1 represents the lower limit of summation and n represents the upper limit
of summation. Note that these numbers can be any integer.
Basically, you will find the sum of the terms that start at the lower
limit and go through the upper limit.
This expression would represent the sum of terms 1 through n:
|
A series is a sum of
terms of a sequence.
It can be written in summation notation  or expanded out in a
sum  .
If you need a review on sequences, feel free to go to Tutorial
54A: Sequences.
|
A finite series is a
series that has a sum of a finite number of terms.
All the examples above are finite series.
|
An infinite series is a
series that has a sum of an infinite number of terms.
In a infinite series, the upper limit is infinity
... which means there is no upper bound.
An example of an infinite series is  or it can be written like  .
The three dots indicate that this pattern will keep continuing on and
on.
|
 Example
1: Find the sum of the series  .
|
You find the terms of the series in the same fashion
that you do for a sequence. Plug the term number in for the given
variable. So in this problem, wherever there is an i in the term, the term
number will replace it.
If you need a review on sequences, feel free to go to Tutorial
54A: Sequences.
The difference between
this problem and a sequence problem is that you will be adding all of
the terms together to get your end result.
What
values will you be plugging in to get the terms that will be summed?
If you said 1 through 5 you are right on!!!
Let’s see what we get when
we add up terms plugging in 1, 2, 3, 4, and 5 for i:
|
Example 2: Find
the sum of the series  .
|
You
find the terms of the series in the same fashion that you do for a
sequence. Plug the term number in for the given variable.
So in this
problem, wherever there is an n in the term, the term number will replace it.
If you need a review on sequences, feel free to go to Tutorial
54A: Sequences.
Also note that this problem has a factorial in it. If you need a
review on factorials, feel free to go to Tutorial 54A: Sequences.
The difference between
this
problem and a sequence problem is that you will be adding all of the
terms together to get your end result.
What
values will you be plugging in to get the terms that will be summed?
If you said 0 through 6 you are right on!!!
Let’s see what we get when
we add up terms plugging in 0, 1, 2, 3, 4, 5, and 6 for n:
|
Example 3: Find
the sum of the series  .
|
You
find the terms of the series in the same fashion that you do for a
sequence. Plug the term number in for the given variable.
So in this
problem, wherever there is an k in the term, the term number will replace it.
If you need a review on sequences, feel free to go to Tutorial 54A: Sequences.
The difference between
this
problem and a sequence problem is that you will be adding all of the
terms together to get your end result.
What
values will you be plugging in to get the terms that will be summed?
If you said 1 through 4 you are right on!!!
Let’s see what we get when
we add up terms plugging in 1, 2, 3, and 4 for k:
Note that the general term is 7, which is a
constant. So no matter what k is, the term is always
7.
Since we were going from 1 to 4, we had 4 terms of 7 be added or 4(7) =
28.
In general, if your general term is constant than the sum will end up
being the number of terms in your series times the constant.
|
Example 4: Write
the series  in summation notation.
Use the index i and let i begin
with 1.
|
Again, this is similar to working with sequences.
You need to analyze what is happening term by term and see how it
relates to the index number.
One thing that is always constant is that the numerator of each term is
1.
However the denominator changes.
We
need to figure out the relationship between i and the denominator:
When i is 1, the denominator is 1
When i is 2, the
denominator is 3.
When i is 3, the
denominator is 9.
When i is 4, the
denominator is 27. |
What
do you think the relationship is?
It looks like the denominator is always 3 to a power. It also
looks like it starts as 3 to the 0 power, then 3 to 1 power, and so
forth. Since we have to start with i = 1, we will have to adjust
the power so that it will start with 0.
So
how will we represent the power?
If you said i - 1,
give yourself a pat on the back.
Putting it together, we can write the summation as  .
An equivalent way to write this is  .
Sometimes you have to play around with it before you get it just
right. You can always check it by putting in the i values and seeing if you
get the given series.
This one does check.
|
Example 5: Write
the series  in summation
notation. Use the index i and let i begin
with 1.
|
Again,
this is similar to working with sequences. You need to analyze
what
is happening term by term and see how it relates to the index
number.
One thing that is always constant is that each term contains an e.
There are also two things that change.
First
let's look at the alternating signs:
The first term is
positive, the second term is negative, the third positive, the fourth
negative and so forth.
When i is odd (1,
3, 5, ...), then the term is positive.
When i is even (2,
4, 6, ...), then the term is negative.
So do you think we are going to have  or  .
If you said you are
correct!!! If n is odd, then this term will be positive. If n is even, then this term
will be negative.
|
Next, let’s look at the exponent on each term.
We
need to figure out the relationship between i and the exponent:
When i is 1, the exponent is 3.
When i is 2, the
exponent is 4.
When i is 3, the exponent is
5.
When i is 4, the exponent is
6.
And so forth.... |
What
do you think the relationship is?
Since we have to start with i = 1, we will have to adjust the power so that it will start with
3.
So
how will we represent the power?
If you said i + 2,
give yourself a pat on the back.
Putting it together, we can write the summation as  .
Sometimes you have to play around with it before you get it just
right. You can always check it by putting in the i values and seeing if you
get the given series.
This one does check.
|
Example 6:
Rewrite the series  using the new index j such that =  .
|
We need to rewrite the series so that they are both
equivalent to each other, but the first one starts at n = 2 and the second one
starts at j = 0.
We
need to find the relationship between n and j and rewrite n in terms of j:
Looking at the lower limits for each, how would you write n in terms of j? What is their
relationship?
If you said n = j + 2, you are
correct!!!
Basically, we need to do a substitution. Wherever we have an n in our general term, we
will replace it with j + 2:
|
Mean of a Sequence
of Numbers
|
When you find a mean of a
set of numbers, you add up all of the numbers and divide it by the
number of values that you have. You are doing the same thing
here. This is just a special set of numbers that come from a
sequence.
|
Example 7: Find
the mean of the sequence  .
|
Note that I will be rounding the square root of 2 to
1.414 and e to
2.718.
We
need to find the mean, so we will be adding all of the values that we
have and dividing that sum by what number?
If you said 5, you are correct!
Let’s see what we get:
|
Practice Problems
These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these
types of problems. Math works just like anything
else, if you want to get good at it, then you need to practice it.
Even the best athletes and musicians had help along the way and lots of
practice, practice, practice, to get good at their sport or instrument.
In fact there is no such thing as too much practice.
To get the most out of these, you should work the problem out on
your own and then check your answer by clicking on the link for the answer/discussion
for that problem. At the link you will find the answer
as well as any steps that went into finding that answer. |
 Practice Problems 1a - 1b: Find
the sum of the given series.
Practice Problem 2a: Write the
series in summation notation. Use the index i and let i begin with 1.
Practice Problem 3a: Rewrite the series
using the new index j as
indicated.
Practice Problem 4a: Find the mean
of the sequence.
Need Extra Help on these Topics?
Last revised on May 17, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.
|
|
