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7 Power Series and Taylor Series Online Quiz Questions

Use this free practice quiz with 20 questions to review 7 Power Series and Taylor Series, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

What is the center of the power series ∑n=0∞cn(x−3)n\sum_{n=0}^{\infty} c_n(x-3)^n?

  1. A

    0

  2. B

    3

  3. C

    -3

  4. D

    13\frac{1}{3}

02
True or false
1 point

True or false: For a power series with a finite radius of convergence, the endpoints must generally be tested separately.

  1. A

    True

  2. B

    False

03
Written response
1 point

What is the radius of convergence of ∑n=1∞(x−4)nn5n\sum_{n=1}^{\infty}\frac{(x-4)^n}{n5^n}?

04
Fill in the blank
1 point

Complete the statement: The geometric series ∑n=0∞rn\sum_{n=0}^{\infty} r^n converges when ∣r∣|r| 1 1.

05
Choose one
1 point

Which power series represents 11+x3\frac{1}{1+x^3} for ∣x∣<1|x|<1?

  1. A

    ∑n=0∞x3n\sum_{n=0}^{\infty}x^{3n}

  2. B

    ∑n=0∞(−1)nxn\sum_{n=0}^{\infty}(-1)^n x^n

  3. C

    ∑n=0∞(−1)nx3n\sum_{n=0}^{\infty}(-1)^n x^{3n}

  4. D

    ∑n=0∞(−x)3n+1\sum_{n=0}^{\infty}(-x)^{3n+1}

06
True or false
1 point

True or false: Differentiating a power series term by term leaves its radius of convergence unchanged, even though endpoint behavior may differ.

  1. A

    True

  2. B

    False

07
Choose all
1 point

Select all correct statements about finding the interval of convergence of a power series.

  1. A

    The ratio test can usually determine the radius.

  2. B

    Each endpoint automatically has the same convergence behavior as the interior.

  3. C

    The endpoints should be substituted into the original series and tested.

  4. D

    A power series with a finite radius necessarily diverges at both endpoints.

08
Written response
1 point

What is the standard name for a Taylor series centered at x=0x=0?

09
Fill in the blank
1 point

The finite sum Pn(x)=∑k=0nf(k)(a)k!(x−a)kP_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k is called the degree-nn .

10
Choose one
1 point

If ∣f(n+1)(t)∣≤M|f^{(n+1)}(t)|\le M on the interval between aa and xx, which expression is the Taylor remainder bound for the degree-nn Taylor polynomial?

  1. A

    Mn!∣x−a∣n\frac{M}{n!}|x-a|^n

  2. B

    M(n+1)!∣x−a∣n+1\frac{M}{(n+1)!}|x-a|^{n+1}

  3. C

    M(n+1)!∣x−a∣n\frac{M}{(n+1)!}|x-a|^n

  4. D

    Mn!∣x−a∣n+1\frac{M}{n!}|x-a|^{n+1}

11
Choose all
1 point

Select all correct statements about the convergence of common Maclaurin series.

  1. A

    The Maclaurin series for exe^x converges for every real xx.

  2. B

    The Maclaurin series for sin⁡x\sin x converges for every real xx.

  3. C

    The geometric series for 11−x\frac{1}{1-x} converges for every real xx.

  4. D

    The Maclaurin series for cos⁡x\cos x has infinite radius of convergence.

12
Open ended
1 point

Find the radius and interval of convergence of ∑n=1∞(x+1)nn2n\sum_{n=1}^{\infty}\frac{(x+1)^n}{n2^n}. Show how you handle both endpoints.

13
Choose one
1 point

Which expression is the degree-5 Maclaurin polynomial for sin⁡x\sin x?

  1. A

    x−x33!x-\frac{x^3}{3!}

  2. B

    x+x33!+x55!x+\frac{x^3}{3!}+\frac{x^5}{5!}

  3. C

    x−x33!+x55!x-\frac{x^3}{3!}+\frac{x^5}{5!}

  4. D

    1+x+x22!+x33!+x44!+x55!1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\frac{x^5}{5!}

14
True or false
1 point

True or false: After a ratio test determines the possible interval of convergence for a power series, both endpoints must generally be tested separately.

  1. A

    True

  2. B

    False

15
Choose one
1 point

A power series has radius of convergence RR. What is the radius of convergence of the series obtained by differentiating it term by term?

  1. A

    The radius becomes 2R2R.

  2. B

    The radius becomes R/2R/2.

  3. C

    The radius remains RR.

  4. D

    The radius becomes infinite.

16
Choose one
1 point

Which expression is the degree-nn Taylor polynomial for ff centered at aa?

  1. A

    Pn(x)=∑k=1nf(k)(a)k!(x−a)k\displaystyle P_n(x)=\sum_{k=1}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k

  2. B

    Pn(x)=∑k=0∞f(k)(a)k!(x−a)k\displaystyle P_n(x)=\sum_{k=0}^{\infty}\frac{f^{(k)}(a)}{k!}(x-a)^k

  3. C

    Pn(x)=∑k=0nf(k)(a)k!(x−a)k\displaystyle P_n(x)=\sum_{k=0}^{n}\frac{f^{(k)}(a)}{k!}(x-a)^k

  4. D

    Pn(x)=f(n)(a)n!(x−a)n\displaystyle P_n(x)=\frac{f^{(n)}(a)}{n!}(x-a)^n

17
Choose one
1 point

What is the interval of convergence of the Maclaurin series ln⁡(1+x)=∑n=1∞(−1)n+1xnn\displaystyle \ln(1+x)=\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^n}{n}?

  1. A

    [−1,1][-1,1]

  2. B

    (−1,1)(-1,1)

  3. C

    (−1,1](-1,1]

  4. D

    [−1,1)[-1,1)

18
Written response
1 point

Using the degree-5 Maclaurin polynomial P5(x)=x−x33!+x55!P_5(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}, enter the approximation to sin⁡(0.2)\sin(0.2) rounded to eight digits after the decimal point.

19
Choose one
1 point

Suppose ∣f(n+1)(t)∣≤M|f^{(n+1)}(t)|\le M between aa and xx. Which expression is a Taylor remainder bound for the degree-nn polynomial?

  1. A

    Mn!∣x−a∣n\displaystyle \frac{M}{n!}|x-a|^n

  2. B

    M(n+1)!∣x−a∣n\displaystyle \frac{M}{(n+1)!}|x-a|^n

  3. C

    Mn!∣x−a∣n+1\displaystyle \frac{M}{n!}|x-a|^{n+1}

  4. D

    M(n+1)!∣x−a∣n+1\displaystyle \frac{M}{(n+1)!}|x-a|^{n+1}

20
Written response
1 point

Using the term-by-term integration result for arctan⁡x\arctan x, what is the coefficient of x7x^7 in its Maclaurin series?