15 Questions
| Total Attempts: 80

Questions and Answers

- 1.Suppose the series S begins with 1 - 0.56 + 0.33 - 0.27 + 0.23 - 0.16 + ... . Also, assume that the terms continue to alternate between positive and negative, and assume that the absolute values of the terms continue to strictly decrease and approach 0. In that case, what are the best upper and lower bounds we can put on the limit of the series, using just the information above?
- 2.EXTRA CREDIT: Evaluate the series
*exactly*. Explain how you did it. (You may skip this extra credit if you want, but you must type something in the box or the site won't let you submit your quiz.) - 3.Let , and let A(x) be an antiderivative of f(x) such that A(0)=0. Warning: A(x) is not an elementary function. Determine the 5th degree Maclaurin polynomial for A(x). Explain your method. (You may use the ^ symbol for powers.)
- 4.Which of these series can be shown to converge by the alternating series test? (Check all correct answer(s); there may be more than one. If the alternating series test doesn't apply, then
*do not*check the box, even if the series does converge.)- A.
1 - 0.9 + 0.89 - 0.889 + 0.8889 - 0.88889 + ...

- B.
1 - 0.1 + 0.01 - 0.001 + 0.0001 - 0.00001 + ...

- C.
1/ln(2) - 1/ln(3) + 1/ln(4) - 1/ln(5) + ...

- D.
1/arctan(1) - 1/arctan(2) + 1/arctan(3) - 1/arctan(4) + ...

- 5.Which of these series can be shown to converge by the ratio test? (Check all correct answer(s); there may be more than one. If the ratio test is inconclusive, then
*do not*check the box, even if the series does converge.)- A.
1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ...

- B.
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ...

- C.
1/2 + 2/4 + 3/8 + 4/16 + 5/32 + 6/64 + ...

- D.
1 + 10/1! + 100/2! + 1000/3! + 10000/4! + ...

- 6.Suppose the series converges when x = -1. Which of the following statements must be true? You may select more than one.
- A.
The series converges when x = 1.

- B.
The series converges when x = 3/4.

- C.
The series diverges when x = -2.

- D.
The series converges absolutely when x = -1/2.

- 7.Suppose the series converges when x=8, but diverges for all x>8. Which of the following
*could be*the interval of convergence of the series? You may select more than one.- A.
(-8,8]

- B.
[-8,8]

- C.
(0,8]

- D.
[0,8]

- 8.What is the radius of convergence of the series ?
- A.
4

- B.
2

- C.
1

- D.
1/2

- E.
None of the above

- 9.If the terms of a series approach 0, then the series must converge.
- A.
True

- B.
False

- 10.If the terms of a series do not approach 0, then the series must diverge.
- A.
True

- B.
False

- 11.Which of the following series can have their terms rearranged (without changing the signs) so that the limit becomes 8888? Check all correct answer(s); there may be more than one.
- A.
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...

- B.
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + ...

- C.
1 - 1/2 + 1/4 - 1/8 + 1/16 - 1/32 + ...

- D.
1 - 2 + 3 - 4 + 5 - 6 + ...

- 12.If the series (with all terms positive) converges, then the series must also converge.
- A.
True

- B.
False

- 13.If a series converges, then we can put 1000 more terms at the beginning (which can be any numbers we want) and it will still have to converge.
- A.
True

- B.
False

- 14.Which of the functions below can be approximated for any x to any accuracy we wish, just by adding enough terms of their Maclaurin series? Check all that apply.
- A.
Cos x

- B.
Sin x

- C.
1/(1-x) (x not equal to 1)

- D.
Arctan x

- E.
E^x

- 15.If , then is equal to:
- A.
1/3

- B.
1/6

- C.
1/12

- D.
1/24

- E.
None of the above