Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Concepts of Convergence and Divergence

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

Example Question #1 : Concepts Of Convergence And Divergence

One of the following infinite series CONVERGES. Which is it?

Possible Answers:

$\sum_{k=1}^{\infty} (-1)^k sin(k)$

$\sum_{k=1}^{\infty} \frac{sin(k)}{k^2}$

$\sum_{k=2}^{\infty} \frac{x}{ln(x)}$

None of the others converge.

$\sum_{k=1}^{\infty} \frac{1}{k}$

Correct answer:

$\sum_{k=1}^{\infty} \frac{sin(k)}{k^2}$

Explanation:

$\sum_{k=1}^{\infty} \frac{sin(k)}{k^2}$ converges due to the comparison test.

We start with the equation $\frac{1}{k^2} =\frac{1}{k^2}$ . Since $sin(k) \leq 1$ for all values of k, we can multiply both side of the equation by the inequality and get $\frac{sin(k)}{k^2} \leq \frac{1}{k^2}$ for all values of k. Since $\sum_{k=1}^{\infty} \frac{1}{k^2}$ is a convergent p-series with p=2 , $\sum_{k=1}^{\infty} \frac{sin(k)}{k^2}$ hence also converges by the comparison test.

Report an Error

Example Question #2 : Concepts Of Convergence And Divergence

Determine the nature of the following series having the general term:

$\frac{9}{(9n+3)(9n+89)}$

Possible Answers:

The series is convergent.

$\frac{9}{82}$

$\frac{3}{9}$

$\frac{1}{9}$

$\frac{2}{9}$

Correct answer:

The series is convergent.

Explanation:

We will use the Limit Comparison Test to show this result.

We first denote the genera term of the series by:

$u_{n}=\frac{9}{(9n+3)(9n+89)}$ and $v_{n}=\frac{9}{n^2}$ .

We have $\lim_{n\rightarrow\infty}\frac{u_n}{v_{n}}=1$ and the series have the same nature .

We know that

$\sum_{n=1}^{\infty} u_{n}=\sum_{n=1}^{\infty} \frac{9}{n^2}$ is convergent by comparing the integral

$9\int_{1}^{\infty} \frac{1}{x^2}dx$ which we know is convergent.

Therefore by the Limit Comparison Test.

we have $\sum_{n=1}^{\infty} v_{n}=\sum_{n=1}^{\infty} \frac{9}{(9n+3)(9n+89)}$ .

Report an Error

Example Question #3 : Concepts Of Convergence And Divergence

If $\sum_{k=1}^{\infty} a_{k}$ converges, which of the following statements must be true?

Possible Answers:

The limit of the $n^{th}$ term as approaches infinity is not zero.

None of the other answers must be true.

The limit of the $n^{th}$ partial sums as approaches infinity is zero.

For some large value of , $a_{k} \geq (a_{k})^{3}$ .

Correct answer:

For some large value of , $a_{k} \geq (a_{k})^{3}$ .

Explanation:

If the series converges, then we know the terms must approach zero. At some point, the terms will be less than 1, meaning when you take the third power of the term, it will be less than the original term.

Other answers are not true for a convergent series by the $n^{th}$ term test for divergence.

In addition, the limit of the $n^{th}$ partial sums refers to the value the series converges to. A convergent series need not converge to zero. The alternating harmonic series is a good counter example to this.

Report an Error

Example Question #2 : Concepts Of Convergence And Divergence

Which of following intervals of convergence cannot exist?

Possible Answers:

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

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

$\small [2,\infty)$

$\small [2,10000]$

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

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

The series diverges to $\small \infty$ .

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

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 diverges.

The series converges.

None of the other answers.

The series conditionally converges.

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 : Series Of Constants

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

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

Possible Answers:

$\frac{3}{4}$

$\frac{4}{3}$

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 #2 : Concepts Of Convergence And Divergence

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:

Diverges

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

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

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 #3 : Concepts Of Convergence And Divergence

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:

Divergent

Inconclusive

Convergent

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 #3 : 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:

Inconclusive

Divergent

Convergent

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

View Calculus 2 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

View Calculus 2 Tutors

Mayowa
Certified Tutor

Logan University, Bachelor of Science, Biology, General.

View Calculus 2 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

All Calculus 2 Resources

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

Popular Subjects

SAT Tutors in Los Angeles, Calculus Tutors in Los Angeles, LSAT Tutors in New York City, ISEE Tutors in Phoenix, Algebra Tutors in Los Angeles, ISEE Tutors in Philadelphia, Algebra Tutors in New York City, French Tutors in Miami, Statistics Tutors in Dallas Fort Worth, GMAT Tutors in Dallas Fort Worth

Popular Courses & Classes

SAT Courses & Classes in Philadelphia, SAT Courses & Classes in Washington DC, ISEE Courses & Classes in Philadelphia, SSAT Courses & Classes in Los Angeles, GRE Courses & Classes in Houston, SAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Atlanta, GMAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

MCAT Test Prep in San Diego, ISEE Test Prep in Philadelphia, GRE Test Prep in Miami, MCAT Test Prep in Los Angeles, MCAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Washington DC, ACT Test Prep in Los Angeles, GMAT Test Prep in Philadelphia, MCAT Test Prep in San Francisco-Bay Area, LSAT Test Prep in Houston

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 : Concepts of Convergence and Divergence

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Concepts Of Convergence And Divergence

Example Question #2 : Concepts Of Convergence And Divergence

Example Question #3 : Concepts Of Convergence And Divergence

Example Question #2 : Concepts Of Convergence And Divergence

Example Question #2 : Concepts Of Convergence And Divergence

Example Question #1 : Concepts Of Convergence And Divergence

Example Question #1 : Series Of Constants

Example Question #2 : Concepts Of Convergence And Divergence

Example Question #3 : Concepts Of Convergence And Divergence

Example Question #3 : Concepts Of Convergence And Divergence

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: