First exam practice sheet
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]
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