## 17Calculus Infinite Series - Where To Start - Choosing A Test

Applications

Tools

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

Calculus Tools

### Articles

How do you know which test to use to determine convergence or divergence of an infinite series? Knowing which test to use is a combination of practice and guessing. However, keep in mind that there may be several tests that will work but there is often a best test that is more efficient and quicker than using other tests.

As you are first learning these tests, here is a hint on how to determine convergence or divergence of a infinite series. Use the tests in this order until one of them tells you that the series converges or diverges:

Group 1. Divergence (nth-Term) Test
Remember that the divergence test can test only for divergence. If the limit goes to zero, you don't know if the series converges or diverges. Said another way, the only two conclusions you can reach from the divergence test are, divergence or inconclusive. If you don't determine divergence, you have to use another test.

Group 2. See if the series matches one of these special forms.
- p-Series
- Geometric Series
- Alternating Series
- Telescoping Series

Group 3. Core Series Tests
- Ratio Test
- Limit Comparison Test or Direct Comparison Test (whichever you find easiest to use)
- Integral Test
- Root Test

The series table is organized in this format.

Once you get some experience, you will find that you can look at a series and determine by what terms are involved, where to start. But this comes only after working many practice problems and deliberate work on your part to discerning patterns. Here are just a few guidelines that may help ( in no special order ). Add more to this list as you learn them.
1. If the series contains a factorial, try the Ratio Test or the Limit Comparison Test.
2. If the series contains a term with n that also has a power of n, try the Root Test.
4. If the series has positive and negative terms, try determining the convergence or divergence of the absolute value of the series and then apply the absolute convergence theorem.
5. If the series looks integrable, try the Integral Test.
6. If you have a polynomial in the denominator that can be factored and no complicated terms in the numerator, try expanding into partial fractions and see if you have a Telescoping Series.

Of course, these guidelines can be used only if you have the option of choosing any test. Make sure you read the problem statement carefully and, if there is any confusion, ask your instructor what they expect.

Here is a video that builds on this discussion. Basically, he has a list of series and he explains his thought process in choosing a test. He doesn't actually work any problems in this video but he does give a great overview on how to look at each series. You will need some actual experience working problems and have some knowledge of all the tests to be able to understand this video.

### PatrickJMT - Infinite Series [12min-46secs]

video by PatrickJMT

You CAN Ace Calculus

### 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

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

Engineering

Circuits

Semiconductors

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.