First exam practice sheet - Austin Community College District



Date of test Tuesday, December 9 in class

Material covered Chapter 8

Allowable materials Calculator (no TI-89 or 92)

3”(5” index card of notes

Trigonometry formulas handout

Sample problems

1. For each of the following sequences, determine whether it converges. If so, find the limit.

a. [pic]

b. [pic]

c. [pic]

2. Find an expression for [pic]and determine whether the sequence converges.

a. [pic]

b. [pic]

3. Without using p-series rule, prove that [pic] diverges.

4. Calculate the sum [pic] within 4 decimal digits of accuracy.

5. For each of the following series, determine whether it converges. If so, find the sum.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

6. State and prove divergence or convergence for each of the following series.

a. [pic]

b. [pic]

c. [pic]

d. [pic]

e. [pic]

f. [pic]

g. [pic]

h. [pic]

i. [pic]

7.

8. Find the radius and interval of convergence of the following power series.

a. [pic][pic]

b. [pic]

c. [pic]

9. For the series [pic], use the integral test remainder approximation to find a value of N that will ensure the error of the approximation [pic] to does not exceed 0.01.

10. Evaluate the indefinite integral [pic] as an infinite series.

11. Use the known MacLaurin series representation of [pic] to answer the following.

a. Estimate the value of [pic] using the 3rd degree Taylor polynomial. Express your answer to 6 decimal places.

b. Use Taylor's inequality to get an upper bound on the error in your computation in part a.

12. Find a MacLaurin series representation of the function [pic]. State the radius and interval of convergence?

13. Find a Taylor series about [pic] for the function [pic]. State the radius and interval of convergence.

14. Use the binomial series to expand the function [pic] as a Maclaurin series. State the radius of convergence.

15. Use the binomial series to expand the function [pic] as a Maclaurin series. State the radius of convergence.

Answers:

1.

a. converges to [pic]

b. diverges (oscillation)

c. diverges (infinite)

2.

a. [pic], converges to 0

b. [pic], diverges

3. Use integral test with [pic]

4. -0.9856

5.

a. converges (geometric), [pic]

b. diverges (telescoping)

c. diverges (geometric)

d. converges (telescoping), [pic]

e. converges (geometric), [pic]

6.

a. diverges, ratio test

b. converges, AST

c. diverges, BCT with [pic]

d. converges, ratio test

e. diverges, ratio test

f. converges, BCT with [pic]

g. diverges, ratio test

h. converges, LCT with [pic]

i. converges, ratio test

7.

a. [pic]

b. [pic]

c. [pic]

8. N=100

9. [pic]

10.

a. 1.213333

b. [pic]

11. [pic], [pic], [pic]

12. [pic], [pic], [pic]

13. [pic], [pic]

14. [pic][pic]

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download