Series (BC Only)

Student Study Session

Series (BC only)

Students should be able to:

Recognize various types of numerical series and efficiently apply the appropriate test.
Determine the sum of an infinite geometric series and be able to use that sum to create a power series and determine its interval of convergence.
Use substitution, differentiation, and integration to create series related to given series.
Use the Ratio Test to determine radius or open interval of convergence of power series.
Use the other tests to check convergence at the endpoints.

Note: Telescoping Series Test questions can be tested using the Alternating Series Test conditions. On recent AP exams the Telescoping Series Test is not listed as an answer choice, therefore it has been omitted in the chart. The Harmonic Series shows up frequently and it is recommended that students know this special case of the p-series test and that it is acceptable to list it as the reason for divergence when it occurs on the AP exam.

Series

Student Study Session

Multiple Choice

1.(calculator not allowed) (1973 BC19)

Which of the following series converge?

II.

III.

(A)I only

(B)III only

(C)I and II only

(D)I and III only

(E)I, II, and III

2.(calculator not allowed) (1985 BC14)

Which of the following series are convergent?

II.

III.

(A)I only

(B)III only

(C)I and III only

(D)II and III only

(E)I, II, and I

3.(calculator not allowed) (1985 BC31)

What are all values of x for which the series converges?

(A)

(B)

(C)

(D)

(E)

4.(calculator not allowed) (1993 BC16)

Which of the following series diverge?

II.

III.

(A)None

(B)II only

(C)III only

(D)I and III only

(E)II and III only

5.(calculator not allowed) (1993 BC27)

The interval of convergence of is

(A)

(B)

(C)

(D)

(E)

6.(calculator not allowed) (1997 BC14)

The sum of the infinite geometric series is

(A)1.60

(B)2.35

(D)2.45

(E)2.50

7.(calculator not allowed) (1997 BC20)

What are allvalues of x for which the series converges?

(A)

(B)

(C)

(D)

(E)

8.(calculator not allowed) (1998 BC18)

Which of the following series converge?

II.

III.

(A)None

(B)II only

(C)III only

(D)I and II only

(E)I and III only

9.(calculator not allowed) (1998 BC22)

If is finite, then which of the following must be true?

(A) converges

(B) diverges

(D) converges

(E) diverges

10.(calculator not allowed) (2003 BC10)

What is the value of ?

(A)1

(B)2

(D)6

(E)The series diverges

11.(calculator not allowed) (2003 BC22)

What are all values of p for which the infinite series converges?

(A)

(B)

(C)

(D)

(E)

12.(calculator not allowed) (2003 BC24)

Which of the following series diverge?

II.

III.

(A)III only

(B)I and II only

(C)I and III only

(D)II and III only

(E)I, II, and III

13.(calculator not allowed) (2008 BC 4)

Consider the series . If the ratio test is applied to the series, which of the following inequalities results, implying that the series converges?

(A)

(B)

(C)

(D)

(E)

14.(calculator not allowed) (2008 BC 12)

Which of the following series converges for all real numbers ?

(A)

(B)

(C)

(D)

(E)

15. (calculator not allowed) (2008 BC16)

What are all vlaues of for which the series converges?

(A)

(B) only

(D) and only

(E) and

16.(calculator allowed) (1998 BC76)

For what integer k, , will both andconverge?

(A)6

(B)5

(D)3

(E)2

17.(calculator allowed) (1998 BC84)

What are all values of x for which the series converges?

(A)

(B)

(C)

(D)

(E)

Free Response

18.(calculator not allowed) 2001 BC6

A function f is defined by

for all x in the interval of convergence of the given power series.

(a)Find the interval of convergence for this power series. Show the work that leads to your answer.

(b)Find .

(c)Write the first three nonzero terms and the general term for an infinite series that represents

(d)Find the sum of the series determined in part (c).

19.(calculator not allowed) 2002 BC6

The Maclaurin series for the function f is given by

on its interval of convergence.

(a)Find the interval of convergence for the Maclaurin series for f. Justify your answer.

(b)Find the first four terms and the general term for the Maclaurin series for .

(c)Use the Maclaurin series you found in part (b) to find the value of .

20.(calculator not allowed) 2006 BC6

The function f is defined by the power series

for all real numbers x for which the series converges. The function g is defined by the power series

for all real numbers x for which the series converges.

(a)Find the interval of convergence of the power series for f. Justify your answer.

(b)The graph of passes through the point . Find and . Determine whethery has a relative minimum, a relative maximum, or neither at . Give a reason for your answer

21.(calculator not allowed) 2008B BC6

Let f be the function given by .

(a)Write the first four nonzero terms and the general term of the Taylor series for f about .

(b)Does the series found in part (a), when evaluated at , converge to ? Explain why or why not.

(c)The derivative of is . Write the first four nonzero terms of the Taylor series

for about .

(d)Use the series found in part (c) to find a rational number A such that .

Justify your answer.

22.(calculator not allowed) 2002B BC6

The Maclaurin series for is with interval of convergence .

(a)Find the Maclaurin series for and determine the interval of convergence.

(bFind the value of .

(c)Give a value of p such that converges, but diverges. Give reasons why your value of p is correct.

(d)Give a value of p such that diverges, but converges. Give reasons why your value of p is correct.

23.(calculator not allowed) 2010B BC6

The Maclaurin series for the function f is given by on its interval of convergence.

(a)Find the interval of convergence for the Macluarin series of f. Justify your answer.

(b)Show that is a solution to the differential equation for , where R is the radius of convergence from part (a).