Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Introduction to Series in Calculus

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 Next →

Example Question #4 : Concepts Of Convergence And Divergence

Which of following intervals of convergence cannot exist?

Possible Answers:

For any $\small q,p$ such that $\small q \leq p$ , the interval [q,p]

$\small [2,10000]$

For any $\small \epsilon>0$ , the interval $\small [a-\epsilon,a+\epsilon]$ for some $\small a\in\mathbb{R}$

$\small [2,\infty)$

Correct answer:

$\small [2,\infty)$

Explanation:

$\small [2,\infty)$ cannot be an interval of convergence because a theorem states that a radius has to be either nonzero and finite, or infinite (which would imply that it has interval of convergence $\small (-\infty,\infty)$ ). Thus, $\small [2,\infty)$ can never be an interval of convergence.

Report an Error

Example Question #5 : Concepts Of Convergence And Divergence

Which of the following statements is true regarding the following infinite series?

$\small \sum_{n=0}^\infty \sin n$

Possible Answers:

The series converges because $\small \lim_{n\to\infty} \sin n=0$

The series diverges, by the divergence test, because the limit of the sequence $\small a_n=\sin n$ does not approach a value as $\small n\to\infty$

The series diverges because $\small \lim_{n\to \infty} \sin n=\alpha$ for some $\small \alpha>0$ and finite.

The series diverges to $\small \infty$ .

Correct answer:

The series diverges, by the divergence test, because the limit of the sequence $\small a_n=\sin n$ does not approach a value as $\small n\to\infty$

Explanation:

The divergence tests states for a series $\small \sum_{n=0}^\infty a_n$ , if $\small \lim a_n$ is either nonzero or does not exist, then the series diverges.

The limit $\small \small \lim_{n\to\infty} \sin n$ does not exist, so therefore the series diverges.

Report an Error

Example Question #1 : Concepts Of Convergence And Divergence

Determine whether the following series converges or diverges:

$\sum_{n=0}^{\infty }\frac{n^3+3n}{2n^3}$

Possible Answers:

The series conditionally converges.

The series diverges.

The series converges.

None of the other answers.

Correct answer:

The series converges.

Explanation:

To prove the series converges, the following must be true:

If $\lim_{n\rightarrow \infty }a_{n}$ converges, then $\sum_{n=0}^{\infty }a_{n}$ converges.

Now, we simply evaluate the limit:

$\lim_{n\rightarrow \infty }\frac{n^3+3n}{2n^3}=\frac{1}{2}$

The shortcut that was used to evaluate the limit as n approaches infinity was that the coefficients of the highest powered term in numerator and denominator were divided.

The limit approaches a number (converges), so the series converges.

Report an Error

Example Question #1 : Geometric Series

Determine whether the following series converges or diverges. If it converges, what does it converge to?

$\sum_{n=0}^{\infty}(2)^{-2n}$

Possible Answers:

$\frac{4}{3}$

$\frac{3}{4}$

Diverges

Correct answer:

$\frac{4}{3}$

Explanation:

First, we reduce the series into a simpler form.

$\sum_{n=0}^{\infty}(2)^{-2n}= \sum_{n=0}^{\infty}(\frac{1}{2})^{2n}= \sum_{n=0}^{\infty}(\frac{1}{4})^{n}$

We know this series converges because

$\lim_{n \rightarrow \infty}(\frac{1}{4})^{n}=0$

By the Geometric Series Theorem, the sum of this series is given by

$\sum_{n=0}^{\infty}(\frac{1}{4})^{n}= \frac{(\frac{1}{4})^0}{1-\frac{1}{4}}= \frac{4}{3}$

Report an Error

Example Question #21 : Introduction To Series In Calculus

Determine whether the following series converges or diverges. If it converges, what does it converge to?

$\sum_{n=1}^{\infty}\frac{1}{n^{-0.5}}$

Possible Answers:

$\frac{1}{1-\sqrt{2}}$

$\frac{\sqrt{2}}{2}$

Diverges

Correct answer:

Diverges

Explanation:

Notice how this series can be rewritten as

$\sum_{n=1}^{\infty}\frac{1}{n^{-0.5}}= \sum_{n=1}^{\infty}\sqrt{n}$

$\lim_{n \rightarrow \infty}\sqrt{n}=\infty$

Therefore this series diverges.

Report an Error

Example Question #1 : Sequences & Series

There are 2 series, $\sum_{n=1}^{\infty}a_{n}$ and $\sum_{n=1}^{\infty}b_{n}$ , and they are both convergent. Is $\sum_{n=1}^{\infty}a_{n} + \sum_{n=1}^{\infty}b_{n}$ convergent, divergent, or inconclusive?

Possible Answers:

Convergent

Divergent

Inconclusive

Correct answer:

Convergent

Explanation:

Infinite series can be added and subtracted with each other.

$\sum_{n=1}^{\infty}a_{n} + \sum_{n=1}^{\infty}b_{n} = \sum_{n=1}^{\infty}( a_{n} + b_{n})$

Since the 2 series are convergent, the sum of the convergent infinite series is also convergent.

Note: The starting value, in this case n=1, must be the same before adding infinite series together.

Report an Error

Example Question #1 : Concepts Of Convergence And Divergence

You have a divergent series $\sum_{n=1}^{\infty} a_{n}$ , and you multiply it by a constant 10. Is the new series $\sum_{n=1}^{\infty}10* a_{n}$ convergent or divergent?

Possible Answers:

Convergent

Inconclusive

Divergent

Correct answer:

Divergent

Explanation:

This is a fundamental property of series.

For any constant c, if $\sum_{n=1}^{\infty} a_{n}$ is convergent then $\sum_{n=1}^{\infty} c*a_{n}$ is convergent, and if $\sum_{n=1}^{\infty} a_{n}$ is divergent, $\sum_{n=1}^{\infty} c* a_{n}$ is divergent.

$\sum_{n=1}^{\infty} a_{n}$ is divergent in the question, and the constant c is 10 in this case, so $\sum_{n=1}^{\infty} 10*a_{n}$ is also divergent.

Report an Error

Example Question #21 : Series In Calculus

There are 2 series, $\sum_{n=1}^{\infty}a_{n}$ and $\sum_{n=1}^{\infty}b_{n}$ , and they are both divergent. Is $\sum_{n=1}^{\infty}a_{n} + \sum_{n=1}^{\infty}b_{n}$ convergent, divergent, or inconclusive?

Possible Answers:

Convergent

Inconclusive

Divergent

Correct answer:

Inconclusive

Explanation:

$\sum_{n=1}^{\infty}a_{n}$ is divergent

$\sum_{n=1}^{\infty}b_{n}$ is divergent

However, unlike convergent series in which the sum of convergent series will produce a convergent series, this is not the case for divergent series. Due to the nature of infinite series, adding together 2 divergent series may be divergent, but it may also produce a convergent series. More information is needed.

$\sum_{n=1}^{\infty}a_{n} + \sum_{n=1}^{\infty}b_{n}$ is inconclusive

Report an Error

Example Question #22 : Series In Calculus

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

$\sum_{n=1}^{\infty}\frac{6^{n+4}}{11^{n}}$

Possible Answers:

Inconclusive

Convergent

Divergent

Correct answer:

Inconclusive

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If $\lim_{n \to \infty} a_{n} \neq 0$ then $a_{n}$ diverges.

However, if $\lim_{n \to \infty} a_{n} = 0$ the test is inconclusive.

Solution:

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty}\frac{6^{n+4}}{11^{n}}$

$\\=\lim_{n \to \infty} 6*\left(\frac{6}{11}\right)^{n} \\ \\= 6*0 \\ = 0$

The series is inconclusive by the divergence test.

Report an Error

Example Question #23 : Series In Calculus

Use the limit test (divergence test) to find if the series is convergent, divergent, or inconclusive.

$\sum_{n=1}^{\infty}\frac{12+6sin(n)}{n}$

Possible Answers:

Convergent

Divergent

Inconclusive

Correct answer:

Inconclusive

Explanation:

Divergence Test and Limit Test are the same tests with different names.

If $\lim_{n \to \infty} a_{n} \neq 0$ then $a_{n}$ diverges.

However, if $\lim_{n \to \infty} a_{n} = 0$ the test is inconclusive.

Solution:

$\lim_{n \to \infty} a_{n} = \lim_{n \to \infty}\frac{12+6sin(n)}{n}$

$\lim_{n \to \infty} \frac{12}{n} + \lim_{n \to \infty} \frac{6sin(n)}{n} = 0 + 0 = 0$

This series is inconclusive by the divergence test.

Report an Error

← Previous 1 2 3 4 Next →

View Calculus 2 Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

View Calculus 2 Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 2 Tutors

Filiberto
Certified Tutor

University of Arizona, Bachelor of Science, Electrical Engineering.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

French Tutors in San Diego, English Tutors in Phoenix, Biology Tutors in Philadelphia, Physics Tutors in Seattle, Calculus Tutors in Dallas Fort Worth, French Tutors in Los Angeles, Biology Tutors in Miami, MCAT Tutors in Phoenix, ACT Tutors in Boston, Statistics Tutors in Denver

Popular Courses & Classes

LSAT Courses & Classes in New York City, LSAT Courses & Classes in Los Angeles, ACT Courses & Classes in Houston, SAT Courses & Classes in Miami, Spanish Courses & Classes in San Francisco-Bay Area, GMAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Washington DC, SAT Courses & Classes in Washington DC, SSAT Courses & Classes in Boston, LSAT Courses & Classes in Seattle

Popular Test Prep

LSAT Test Prep in Atlanta, ACT Test Prep in Houston, ISEE Test Prep in Miami, ISEE Test Prep in Atlanta, LSAT Test Prep in Washington DC, LSAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Boston, ISEE Test Prep in Los Angeles, ACT Test Prep in New York City, LSAT Test Prep in New York City

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 2 : Introduction to Series in Calculus

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #4 : Concepts Of Convergence And Divergence

Example Question #5 : Concepts Of Convergence And Divergence

Example Question #1 : Concepts Of Convergence And Divergence

Example Question #1 : Geometric Series

Example Question #21 : Introduction To Series In Calculus

Example Question #1 : Sequences & Series

Example Question #1 : Concepts Of Convergence And Divergence

Example Question #21 : Series In Calculus

Example Question #22 : Series In Calculus

Example Question #23 : Series In Calculus

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: