College Algebra
Tutorial 54B: Series
Learning Objectives
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
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:
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:
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.
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:
Practice Problems
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?
There were no good websites found to help us with the topics on this page.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on May 17, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.