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

17Calculus - Infinite Series


Using Limits



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Other Applications


Improper Integrals

Trig Integrals


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Infinite Series





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Infinite Series is an unusual calculus topic but series can be very useful for computation and problem solving, especially when it comes to integration and differential equations. However, in the realm of infinity, unusual things start to happen. So more care is required. Also, when you start working with infinite series, the question of convergence becomes important.

Beware, this topic will challenge you. Go to the infinite series study techniques page for how to effectively study and learn this topic.

This page covers the introduction to infinite series and contains some basic information to get you started.

In A Hurry or Ready To Review For Your Exam?

If you have your exam tomorrow and you need to learn Infinite Series in a hurry, we recommend a video with 100 practice problems. We have built a page around that video with each problem stated for you and a video clip with the solution.
NOTE: This is NOT the best way to learn Infinite Series and ace your exam. However, if you are pressed for time, this video will give you plenty of practice.
This video will also help you review for your exam once you have studied each individual topic.

Getting Started

To get started, let's watch some videos. Because of the complexity of this topic, it is best to watch all of these introduction videos. There is some repetition but the information comes with different viewpoints and it will help you to assimiliate the information if you watch all of them.

This first video is a very basic introduction to series. If you have some experience with series and know how to expand and use the terms, you can skip this video.

Khan Academy - Sequences and Series (part 1) [9min-48secs]

In this video he introduces sequences, series and sigma notation. He does a pretty good job explaining the basics required for understanding series.

About 3 minutes into the video, he says something that is correct in this context but not in general. He says that you can add numbers in any order and get the same answer. This is true for a finite list of numbers, like he is doing here, but it is NOT always true if you have an infinite list of numbers. You will probably run across this case in your textbook while you are studying this topic.

video by Khan Academy

This second video contains a quick introduction to series and uses the geometric series to show some examples.

PatrickJMT - What is a Series? [12min-23secs]

video by PatrickJMT

This next video is the most important introduction video. If you watch only one, this is the one to watch. This video gives a great introduction and overview of many of the series tests.

Dr Chris Tisdell - Intro to series + the integral test [47min-36secs]

video by Dr Chris Tisdell

This video contains important theory that will help you get a good grasp of series. He uses improper integrals to explain the theory of series and introduce the ideas of convergence and divergence. The presenter is clear, communicates well and is interesting to listen to.

MIT OCW - Lec 37 | MIT 18.01 Single Variable Calculus, Fall 2007 [50min-11secs]

video by MIT OCW

This last introduction video is really good to watch and contains a visual example of what he calls a visceral example of infinity. It will help you (a lot) to get a feel for series. The presenter is the same person as the previous video (this is actually a continuation of the previous video) and is quite interesting. This video is recommended but not required for the rest of your work with infinite series.

MIT OCW - Lec 38 | MIT 18.01 Single Variable Calculus, Fall 2007 [21min-34secs]

video by MIT OCW

Okay, now that you have a grounding on the basics, let's get down to some details. We suggest strongly suggest that you read the infinite series study techniques page next. Then either the basic properties page or the sequences page is your next logical step.

infinite series 17calculus youtube playlist

Here is another excellent playlist from Michel vanBiezen.

Michel vanBiezen - sequences and series youtube playlist

You CAN Ace Calculus

Topics You Need To Understand For This Page

Related Topics and Links

related topics on other pages

list of infinite series practice problems

external links you may find helpful

Pauls Online Notes - Series - The Basics

Wikipedia - Series

User Comments

Hi! I recently found the 17calculus website and shared it with a friend and we have been screaming about it for a while because it helps immensely. We really appreciate how easy it is to follow, especially with the breakdown of questions and answers and practice exams. We have a infinite series test tomorrow morning and all of the explanations of which test to use, not to use, and why are super helpful and really make us confident about how we'll do tomorrow. Thank you to the 17calculus team for this great resource!! It's a great addition to what I do in class and it really has made me confident for my test tomorrow.

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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