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BC 2,3 Problem Set #4

(Due Thursday, November 15)

Name: ______________

*Please show appropriate work – no calculator or computers – except to check work done by hand. Work should be shown clearly, using correct mathematical notation. Please show enough work on all problems (unless specified otherwise) so that others could follow your work and do a similar problem without help. Collaboration is encouraged, but in the end, the work should be your own. *

2011 AP CALCULUS #6

1.

Let

*x*

3

(a) Use the Maclaurin series for

*x*

to write the first four nonzero terms and the general term of the

Maclaurin series for

*f*

.

(b) The radius of convergence of the Maclaurin series for

*f *

is 1. Determine the interval of convergence. Show the work that leads to your answer.

(c) Write the first four nonzero terms of the Maclaurin series for

*f t*

2

( ).

If

*x*

0

2

*f t dt*

, use the f irst two nonzero terms of the Maclaurin series for

*g *

to approximate

*g*

(

1

)

.

(d) The Maclaurin series for

*g*

, evaluated at

*x *

= 1, is a convergent alternating series with individual terms that decrease in absolute value to 0. Show that your approximation in part (c) must differ from

*g*

(

1

) by less than

1

5

.

BC 2,3 Problem Set #4

(Due Thursday, November 15)

Name: ______________

2.

Let

*a n*

be a decreasing sequence of positive numbers. Suppose that the alternating series

*n*

0 conditionally convergent. Now define

*b n*

*n a n*

for

*n*

0

.

*n a n*

is

(a) Show that

*n*

0

*b*

3

*n*

converges.

(b) Determine whether

*n*

0

*b*

3

*n*

is conditionally convergent or absolutely convergent? Justify your answer.

BC 2,3 Problem Set #4

(Due Thursday, November 15)

Name: ______________

3.

Find the value of

0

1

*x*

2

*e dx*

with an error of at most .001. Show the error analysis clearly. You may use your calculator to do any calculations here, but the analysis should be clear.

4.

Determine the limit:

*n*

lim

0 sin

*n n*

1

*n*

2

BC 2,3

5.

Let

*k*

2

*k n*

1

*k*

!

. Calculate

*n*

2

Problem Set #4

(Due Thursday, November 15)

Name: ______________