Series - Manasquan Public Schools

[Pages:3]Series

Non-Calculator Problems

(1) What are all values of x for which the series x - x2 + x3 - x4 + ... + (-1)n+1 xn + ... converges?

234

n

(a) -1 x 1 (b) -1 x < 1

(c) -1 < x 1

(d) -1 < x < 1

(e)

(2) (-1)n ( )2n = n=0 (2n)!

(a) 1

(b) -1

(c)

(d) 2

(e) e

Calculator Problems

(3) The Maclaurin series for a function f is given by

xn

. What is the value of

f (4) (0) , the fourth

n=0 2n

derivative of f at x = 0 ?

(a) 1

(b) 2

(c) 3

(d) 4

(e) 5

(4) Let f be a function whose seventh derivative is f (7) (x) = 10, 000 cos x . If x = 1 is in the interval

of convergence of the power series for this function, the Taylor polynomial of degree six centered

at x = 0 will approximate f (1) with an error of not more than

(a) 2.45 x 10-5

(b) 1.98 x 10-4

(c) 3.21 x 10-2

(d) 0.248

(e) 1.984

(5) Let f (x) be a function whose Taylor series converges for all x. If f (n) (x) < 1where f (n) (x) is

the nth derivative of f (x) , what is the minimum number of terms of the Taylor series centered

at x = 1 necessary to approximate f (1.2) with an error of less than 0.00001 ?

(a) Three (b) Four

(c) Five

(d) Six

(e) Ten

 (6)

What are all values of p for which

1 1 x p

dx

converges?

(a) p > 0

(b)

p

>

1

(c) p > 1

(d) p >

(e) There is no value of p for which the integral converges

(7)

Let

T (x)

=

k =0

1 2

k

(x - 3)k k!

be the Taylor series for a function f. What is the value of

f (10) (3) ,

the tenth derivative of f at x = 3 ?

(a) 5.382 x10-10 (d) 4.883 x10-4

(b) 2.691 x10-10 (e) 1.953 x10-3

(c) 9.766 x10-4

(8)

What

is

the

approximate

value

of

cos

1 2

obtained

by

using

a

fourth-degree

Taylor

Polynomial

for cos x centered at x = 0 ?

(a) 1 - 1 + 1 2 48 3840

(d) 1- 1 + 1 8 64

(b) 1 - 1 + 1 2 24 640

(e) 1- 1 + 1 8 384

(c) 1- 1 + 1 4 16

2005 BC6 2007 BC6 Form B

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