Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Maclaurin Series

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 : Maclaurin Series

Suppose that $\small f(x)=\cos x^3$ . Calculate $\small f^{(48)}(0)$ .

Possible Answers:

$\small \frac{48!}{12!}$

$\small 1$

$\small \frac{48!}{16!}$

Correct answer:

$\small \frac{48!}{16!}$

Explanation:

Let's find the power series of $\small \cos x^3$ centered at $\small x=0$ to find $\small f^{(48)}(0)$ . We have

$\small \small \small \small \cos x^3= \sum_{n=0}^\infty \frac{(-1)^n(x^3)^{2n}}{(2n)!}=\sum_{n=0}^\infty \frac{(-1)^nx^{6n}}{(2n)!}=1-\frac{x^6}{2!}+\frac{x^{12}}{4!}-...$

This series is much easier to differentiate than the expression $\small \cos x^3$ . We must look at term $\small x^{48}$ , which is the only constant term left after differentiating 48 times. This is the only important term, because when we plug in $\small x=0$ , all of the non-constant terms are zero. So we must have

$\small \small (\cos x^3)^{(48)}|_{x=0}= \left( \frac{x^{48}}{16!}\right )^{(48)}|_{x=0}=\frac{48!}{16!}$

Report an Error

Example Question #1 : Maclaurin Series

What is the value of the following infinite series?

$\small \sum_{n=0}^{\infty} \frac{(-1)^n\pi^{2n+1}}{3^{2n+1}(2n+1)!}$

Possible Answers:

$\small \frac{\sqrt{3}}{2}$

$\small \arctan{\pi/3}$

$\small \frac{1}{2}$

$\small \pi/4$

Correct answer:

$\small \frac{\sqrt{3}}{2}$

Explanation:

We can recognize this series as $\small \sin(x)$ since the power series is

$\small \sin x=\sum_{n=0}^{\infty} \frac{(-1)^nx^{2n+1}}{(2n+1)!}$

with the value $\small \frac{\pi}{3}$ plugged into $\small x$ since

$\small \small \sum_{n=0}^{\infty} \frac{(-1)^n\pi^{2n+1}}{3^{2n+1}(2n+1)!}=\sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+1)!}\left(\frac{\pi}{3} \right )^{2n+1}$ .

So then we have

$\small \small \sum_{n=0}^{\infty} \frac{(-1)^n\pi^{2n+1}}{3^{2n+1}(2n+1)!}=\sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}$ .

Report an Error

Example Question #3 : Maclaurin Series

What is the value of the following infinite series?

$\small \small \small \sum_{n=0}^\infty \frac{1+(-1)^n}{n!}$

Possible Answers:

$\small \frac{1}{2}(e+e^{-1})$

$\small e+e^{-1}$

The infinite series diverges.

$\small e^2+e^{-2}$

$\small e+e^{1/2}$

Correct answer:

$\small e+e^{-1}$

Explanation:

The infinite series can be computed easily by splitting up the two components of the numerator:

$\small \small \small \small \sum_{n=0}^\infty \frac{1+(-1)^n}{n!}=\sum_{n=0}^\infty \frac{1}{n!}+\sum_{n=0}^\infty \frac{(-1)^n}{n!}$

Now we recall the MacLaurin series for the exponential function $\small e^x$ , which is

$\small e^x=\sum_{n=0}^\infty \frac{x^n}{n!}$

which converges for all $\small x$ . We can see that the two infinite series are $\small e^x$ with $\small x=1,-1$ , respectively. So we have

$\small \small \small \small \small \sum_{n=0}^\infty \frac{1+(-1)^n}{n!}=\sum_{n=0}^\infty \frac{1}{n!}+\sum_{n=0}^\infty \frac{(-1)^n}{n!}=e^1+e^{-1}$

Report an Error

Example Question #4 : Maclaurin Series

Find the value of the infinite series.

$\small \sum_{n=0}^{\infty} \frac{\pi^{4n+3}(-1)^n}{(2n+1)}$

Possible Answers:

$\small \pi \sin \pi^2$

$\small \pi \cos \pi^4$

The series does not converge.

$\small \cos \pi^3$

$\small -\pi\cos \pi^2$

Correct answer:

$\small \pi \sin \pi^2$

Explanation:

We can evaluate the series

$\small \sum_{n=0}^{\infty} \frac{\pi^{4n+3}(-1)^n}{(2n+1)}$

by recognizing it as a power series of a known function with a value plugged in for $\small x$ . In particular, it looks similar to $\small \sin x$ :

$\small \sin x =\sum_{n=0}^{\infty}\frac{(-1)^n x^{2n+1}}{(2n+1)!}$

After manipulating the series, we get

$\small \small \sum_{n=0}^{\infty} \frac{\pi^{4n+3}(-1)^n}{(2n+1)}=\pi\sum_{n=0}^{\infty} \frac{\pi^{4n+2}(-1)^n}{(2n+1)}=\pi \sum_{n=0}^{\infty} \frac{(\pi^2)^{2n+1}(-1)^n}{(2n+1)}$ .

Now it suffices to evalute $\small \sum_{n=0}^{\infty} \frac{(\pi^2)^{2n+1}(-1)^n}{(2n+1)}$ , which is $\small \sin \pi^2$ .

So the infinite series has value

$\small \small \small \sum_{n=0}^{\infty} \frac{\pi^{4n+3}(-1)^n}{(2n+1)}=\pi \sum_{n=0}^{\infty} \frac{(\pi^2)^{2n+1}(-1)^n}{(2n+1)}=\pi \sin \pi^2$ .

Report an Error

Example Question #5 : Maclaurin Series

Find the value of the following infinite series:

$\small \sum_{n=0}^\infty \frac{\pi^{2n+1}(-1)^n}{6^{2n+1}(2n+1)!}$

Possible Answers:

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

$\small \frac{\sqrt{3}}{2}$

$\small -\frac{\sqrt{3}}{2}$

$\small \frac{1}{2}$

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

Correct answer:

$\small \frac{1}{2}$

Explanation:

After doing the following manipulation:

$\small \small \sum_{n=0}^\infty \frac{\pi^{2n+1}(-1)^n}{6^{2n+1}(2n+1)!}=\sum_{n=0}^\infty \left(\frac{\pi}{6} \right )^{2n+1}\frac{(-1)^n}{(2n+1)!}$

We can see that this is the power series

$\small \sin x=\sum_{n=0}^\infty \frac{(-1)^nx^{2n+1}}{(2n+1)!}$ with $\small \small x=\frac{\pi}{6}$ plugged in.

So we have

$\small \small \small \sum_{n=0}^\infty \frac{\pi^{2n+1}(-1)^n}{6^{2n+1}(2n+1)!}=\sin \frac{\pi}{6}=\frac{1}{2}$

Report an Error

Example Question #6 : Maclaurin Series

Find the value of the following series.

$\small \sum_{k=0}^\infty\frac{2^k+(-1)^k}{k!}$

Possible Answers:

$\small e^{-2}+e^{-1}$

Divergent.

$\small e^2+e^{-1}$

$\small e^{1/2}-e^{-1}$

$\small e^2-e^{-1}$

Correct answer:

$\small e^2+e^{-1}$

Explanation:

We can split up the sum to get

$\small \small \sum_{k=0}^\infty\frac{2^k+(-1)^k}{k!}=\sum_{k=0}^\infty\frac{2^k}{k!}+\sum_{k=0}^\infty\frac{(-1)^k}{k!}$ .

We know that the power series for $\small e^x$ is

$\small \sum_{n=0}^\infty \frac{x^n}{n!}$

and that each sum,

$\small \sum_{k=0}^\infty\frac{2^k}{k!}$

and

$\small \sum_{k=0}^\infty\frac{(-1)^k}{k!}$

are simply $\small e^{x}$ with $\small x=2,-1$ plugged in, respectively.

Thus,

$\small \small \small \sum_{k=0}^\infty\frac{2^k+(-1)^k}{k!}=\sum_{k=0}^\infty\frac{2^k}{k!}+\sum_{k=0}^\infty\frac{(-1)^k}{k!}=e^2+e^{-1}$ .

Report an Error

Example Question #7 : Maclaurin Series

Find the value of the infinite series.

$\small \sum_{n=1}^\infty \frac{(\ln 5)^n}{n!}$

Possible Answers:

Infinite series does not converge.

$\small \sqrt{5}$

$\small 4$

$\small 5$

$\small \ln 5$

Correct answer:

$\small 4$

Explanation:

The series

$\small \sum_{n=1}^\infty \frac{(\ln 5)^n}{n!}$ looks similar to the series for $\small e^x$ , which is $\small e^x=\sum_{n=0}^{\infty} \frac{x^n}{n!}$

but the series we want to simplify starts at $\small n=1$ , so we can fix this by adding a $\small 1$ and subtracting a $\small 1$ , to leave the value unchanged, i.e.,

$\small \small \sum_{n=1}^\infty \frac{(\ln 5)^n}{n!}=-1+1+\sum_{n=1}^\infty \frac{(\ln 5)^n}{n!}=-1+\sum_{n=0}^\infty \frac{(\ln 5)^n}{n!}$ .

So now we have $\small e^x$ with $\small x=\ln 5$ , which gives us $\small e^{\ln 5}=5$ .

So then we have:

$\small \small \sum_{n=1}^\infty \frac{(\ln 5)^n}{n!}=-1+\sum_{n=0}^\infty \frac{(\ln 5)^n}{n!}=-1+5=4$

Report an Error

Example Question #1 : Maclaurin Series

Write out the first two terms of the Maclaurin series of the following function:

f(x)=x^3+2x^2+1

Possible Answers:

0-1

1+0

0+0

0+1

Correct answer:

1+0

Explanation:

The Maclaurin series of a function is simply the Taylor series of a function, but about x=0 (so a=0 in the formula):

$\sum_{n=0}^{\infty }\frac{f^{n}(a)(x-a)^{n}}{n!}$

To write out the first two terms (n=0 and n=1), we must find the first derivative of the function (because the zeroth derivative is the function itself):

f'(x)=3x^2+4x

The derivative was found using the following rule:

$\frac{d}{dx}(x^n)=nx^{n-1}$

Next, use the general form, plugging in n=0 for the first term and n=1 for the second term:

$\frac{(0 + 0+1)(x)^0}{0!}+\frac{(0+0)(x^1)}{1!}=1+0$

Report an Error

Example Question #3091 : Calculus Ii

Find the Maclaurin series for the function:

$\frac{1}{e^x}$

Possible Answers:

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

$\sum_{n=0}^{\infty}\frac{x^n}{n!}\:ln(n)$

$\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!\:ln(n)}$

$\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!}$

$\sum_{n=0}^{\infty}(-1)^nx^n$

Correct answer:

$\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!}$

Explanation:

Write Maclaurin series generated by a function f. The Maclaurin series is centered at a=0 for the Taylor series.

$\sum_{b=0}^{\infty}\frac{f^b(a=0)}{b!}x^b=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+...$

Evaluate the function and the derivatives of $f(x)=\frac{1}{e^x}$ at x=0 .

$f(x=0)=e^{-x}=e^0=1$

$f'(x=0)= -e^{-x}= -\frac{1}{e^0}= -1$

$f''(x=0)=e^{-x}=e^0=1$

Substitute the values into the power series. The series pattern can be seen as alternating and increasing order.

$1-x+\frac{x^2}{2!}-\frac{x^3}{3!}+\frac{x^4}{4!}... =\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!}$

Report an Error

Example Question #10 : Maclaurin Series

Find the first three terms of the Maclaurin series for the following function:

$f(x)=\cos(x)+x^2$

Possible Answers:

$1+0+\frac{x^2}{4}$

$0+0+\frac{1}{2}$

0+0+0

1+0

Correct answer:

$1+0+\frac{x^2}{4}$

Explanation:

The Maclaurin series of a function is simply the Taylor series for the function about a=0:

$\sum_{n=0}^{\infty }f^{(n)}(a)\frac{(x-a)^n}{n!}$

First, we can find the zeroth, first, and second derivatives of the function (n=0, 1, and 2 are the first three terms).

f^0=cos(0)+0^2=1

f^1=-sin(0)+2(0)=0

f^2=-cos(0)+2=1

Plugging these values into the formula we get the following.

$\sum_{n=0}^{\infty }f^{(n)}(a)\frac{(x-a)^n}{n!}$

$f^0(0)+f^1(0)\frac{(x-0)^1}{1!}+f^2(0)\frac{(x-0)^2}{2!}$

$1+0+\frac{x^2}{4}$

Report an Error

View Calculus 2 Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

View Calculus 2 Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Calculus 2 Tutors

Paul
Certified Tutor

University of Phoenix-Albuquerque Campus, Bachelor of Science, Computer and Information Systems Security.

All Calculus 2 Resources

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

Popular Subjects

Statistics Tutors in Los Angeles, LSAT Tutors in Philadelphia, Biology Tutors in Atlanta, Reading Tutors in Seattle, Biology Tutors in Denver, Reading Tutors in New York City, French Tutors in Seattle, Chemistry Tutors in Dallas Fort Worth, LSAT Tutors in Seattle, Algebra Tutors in Philadelphia

Popular Courses & Classes

SAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Miami, Spanish Courses & Classes in New York City, GRE Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Atlanta, SSAT Courses & Classes in Phoenix, SAT Courses & Classes in Chicago, ISEE Courses & Classes in Atlanta, SSAT Courses & Classes in Philadelphia

Popular Test Prep

GRE Test Prep in Atlanta, SSAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Boston, GRE Test Prep in Miami, GRE Test Prep in New York City, SAT Test Prep in Philadelphia, GMAT Test Prep in San Diego, MCAT Test Prep in Houston, ISEE Test Prep in Philadelphia, GMAT Test Prep in Seattle

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 : Maclaurin Series

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Maclaurin Series

Example Question #1 : Maclaurin Series

Example Question #3 : Maclaurin Series

Example Question #4 : Maclaurin Series

Example Question #5 : Maclaurin Series

Example Question #6 : Maclaurin Series

Example Question #7 : Maclaurin Series

Example Question #1 : Maclaurin Series

Example Question #3091 : Calculus Ii

Example Question #10 : Maclaurin Series

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: