\( \newcommand{\abs}[1]{\left| \, {#1} \, \right| } \) \( \newcommand{\cm}{\mathrm{cm} } \) \( \newcommand{\sec}{ \, \mathrm{sec} \, } \) \( \newcommand{\vhat}[1]{\,\hat{#1}} \) \( \newcommand{\vhati}{\,\hat{i}} \) \( \newcommand{\vhatj}{\,\hat{j}} \) \( \newcommand{\vhatk}{\,\hat{k}} \) \( \newcommand{\vect}[1]{\boldsymbol{\vec{#1}}} \) \( \newcommand{\norm}[1]{\|{#1}\|} \) \( \newcommand{\arccot}{ \, \mathrm{arccot} \, } \) \( \newcommand{\arcsec}{ \, \mathrm{arcsec} \, } \) \( \newcommand{\arccsc}{ \, \mathrm{arccsc} \, } \) \( \newcommand{\sech}{ \, \mathrm{sech} \, } \) \( \newcommand{\csch}{ \, \mathrm{csch} \, } \) \( \newcommand{\arcsinh}{ \, \mathrm{arcsinh} \, } \) \( \newcommand{\arccosh}{ \, \mathrm{arccosh} \, } \) \( \newcommand{\arctanh}{ \, \mathrm{arctanh} \, } \) \( \newcommand{\arccoth}{ \, \mathrm{arccoth} \, } \) \( \newcommand{\arcsech}{ \, \mathrm{arcsech} \, } \) \( \newcommand{\arccsch}{ \, \mathrm{arccsch} \, } \)

17Calculus Infinite Series - Basic Properties

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This page covers some of the basic properties of Infinite Series that you need to know in order to use the series tests.

First, let's look get a big picture of series and what you can expect to see.

Infinite Series Big Picture

The study of Infinite Series is perhaps the most difficult material of all of calculus. It will help you to have a big picture. The study of series can be divided into 3 parts.

Part 1 - Introduction

Section 1: Sequences
This is the only section on sequences and they are introduced as an underlying structure of series.

Part 2 - Special Series and Series Tests

These topics will probably appear in a different order in your textbook. They are grouped here by type. As you learn the series in this section, notice that we are looking at series that contain only numbers and indices. So, if the series converges, it will converge to a number. Sometimes we know this number or can find it. More often, though, we don't know what the number is. We are interested only in whether the series converges or diverges.
Hint: Get to know the Ratio Test very well. You will be using it a lot in Part 3.
Section 2: Series and Convergence
This section usually introduces the series and their relationship to sequences, as well as the idea of convergence.
Section 3: p-Series
This section begins the discussion of specific types of series and under what conditions they converge and diverge. Knowing these is important since you will use them later to determine convergence or divergence of series that are not in a special form.
Section 4: Geometric Series
This is an extremely important series that is used extensively when working with infinite series, Taylor Series and Power Series.
Section 5: Telescoping Series
This is a special series the doesn't show up much but you can learn a lot about how series work when you study this type of series. Also, if you recognize that your series is a telescoping series, it is easy to determine what it converges to.
Section 6: Alternating Series
This is a unique type of series and begins the the discussion of series tests.
Section 7: Ratio Test
Of all the tests, this is, by far, the most important and most used. It is critical that you understand this series and how to use it. Also, my experience has shown that often textbooks will combine this test in the same section with the Root Test, which is the least used test.
Section 8: Series Comparisons
This section discusses two important comparison tests, the Limit Comparison Test and the Direct Comparison Test. These are very important tests but require a bit of study to get your head around them (especially the Direct Comparison Test).
Section 9: Integral Test
This is a fairly useful test but, as you know from your study of integration, it is not always possible to integrate functions. So I usually try another test before using this one.
Section 10: Root Test
Finally, this test has limited usefulness but can come in handy under certain conditions.

Part 3 - Applications of Series

In this third part, notice that we are now working with series that contain a variable (usually x). So we will end up with a function, not a number, if we know what the series converges to. This is an important distinction that may not be obvious when first learning these sections.
Section 11: Taylor Series
Sometimes this material is broken into two additional subsections, Taylor Polynomials (finite series) and Taylor Series (infinite series).
Section 12: Power Series
This is a great section for representing functions as Power Series (Geometric Series).

In both Sections 11 and 12, you will probably find some discussion about approximating functions and series and the resulting error. This material looks harder than it is. So read the material in this app and see if it helps you.

Infinite Series Properties

When working with infinite series, it is not always possible to determine what a series converges to, if it converges. It is enough to know whether it converges or diverges. Here are some important properties of series that will help you as you work with them.

For the following properties, we will assume that we have two convergent series \(\sum{a_n}\), which converges to \(A\) and \(\sum{b_n}\), which converges to \(B\). We also have a real number \(c\).

1. The series \(\sum{[c\cdot a_n]}\) converges and \(\sum{[c \cdot a_n]} = c\sum{a_n} = cA\) .

2. The series \(\sum{[a_n \pm b_n]}\) converges and \(\sum{[a_n \pm b_n]} = \) \(\sum{a_n} \pm \sum{b_n} =\) \(A \pm B\)

3. Adding or subtracting a finite number of real numbers to a series does not change whether or not it converges or diverges.

4. If we have a divergent series \(\sum{d_n}\), \(\sum{a_n} + \sum{d_n}\) diverges.

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Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

\(\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }\)

\(\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }\)

\(\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }\)

\(\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }\)

Set 2 - squared identities

\( \sin^2t + \cos^2t = 1\)

\( 1 + \tan^2t = \sec^2t\)

\( 1 + \cot^2t = \csc^2t\)

Set 3 - double-angle formulas

\( \sin(2t) = 2\sin(t)\cos(t)\)

\(\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }\)

Set 4 - half-angle formulas

\(\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }\)

\(\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }\)

Trig Derivatives

\(\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }\)

 

\(\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }\)

\(\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }\)

 

\(\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }\)

\(\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }\)

 

\(\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }\)

Inverse Trig Derivatives

\(\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }\)

 

\(\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }\)

\(\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }\)

 

\(\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }\)

\(\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

 

\(\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }\)

Trig Integrals

\(\int{\sin(x)~dx} = -\cos(x)+C\)

 

\(\int{\cos(x)~dx} = \sin(x)+C\)

\(\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C\)

 

\(\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C\)

\(\int{\sec(x)~dx} = \) \( \ln\abs{\sec(x)+\tan(x)}+C\)

 

\(\int{\csc(x)~dx} = \) \( -\ln\abs{\csc(x)+\cot(x)}+C\)

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