AP Calculus (BC) Chapter 9 Test No Calculator Section - GitHub Pages

WORKSHEET: Series, Taylor Series

AP Calculus (BC) Chapter 9 Test No Calculator Section

ap-calc.github.io

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Date:

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1

Part I. Multiple-Choice Questions (5 points each; please circle the correct answer.)

(-1)nen

1. The series an =

n2 diverges because

n=1

n=1

I. The terms do not tend to 0 as n tends to .

II. The terms are not all positive. III. lim an+1 > 1.

n an

(A) I only (B) II only (C) III only (D) I and II only (E) I and III only

(3x + 2)n+1

2. The interval of convergence for the series

n5/2

is

n=1

1 (A) -1 x < -

3 1 (B) -1 < x - 3 1 (C) -1 x - 3 1 (D) x 1 3 1 (E) -1 < x < 3

n(x - a)n

3. Given that f (x) =

2n on the interval of convergence of the Taylor series,

n=0

f (4)(a) =

(A) 0

(B) 6

(C) 9 1

(D) 4 1

(E) 4!

2

4. Which of the following series converge?

n2 - n + 5

I.

n7/2 + 1 .

n=1

(-1)n3

II.

n

n=1

cos 2n

III.

n2

.

n=1

(A) I and II only (B) I and III only (C) II and III only (D) They all do! (E) None of them do!

5. 1 - 2 + 4 + ? ? ? + (-1)n 2n + ? ? ? =

2! 4!

(2n)!

(A) 0 (B) -1 (C) (D) 1 (E) -

3

Part II. Free-Response Questions

1. A function f is defined by

f (x)

=

1 +

2 x+

3 x2 + ? ? ? + n + 1 xn + ? ? ?

4 42 43

4n+1

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

(a) (4 points) Find the interval of convergence for this power series. Show the work that leads to your answer.

(b)

(3 points) Find lim x0

f (x) - x

1

4.

4

#1, continued; f (x)

=

1 +

2 x+

3 x2 + ? ? ? + n + 1 xn + ? ? ?

4 42 43

4n+1

(c) (3 points) Write the first three nonzero terms and the general term for an infinite

2

series that represents f (x) dx.

0

(d) (4 points) Find the sum of the series determined in part (c). 5

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