CALCULUS BCREVIEW SHEET 3 ON SERIES

Work the following on notebook paper. Use your calculator only on 4(c), 10, 15, and 16.

find the radius and interval of convergence.

1. Find the radius and interval of convergence of

________________________________________________________________________________

2. (a) Find the interval of convergence for

(b) Write the first four nonzero terms and the general term for , and find its

interval of convergence.

3. (a) Find a power series for centered at x = 0. Write the first four

nonzero terms and the general term. (b) Use your answer to (a) to find the first four nonzero terms and the general term for

(c) Use your answer to (b) to approximate , using Justify your answer.

_____________________________________________________________________________For problems 5 – 8, write the first four nonzero terms and the general term.

4. Maclaurin series for

5. Power series for centered at x = 0

7. Suppose is approximated near x = 0 by a fifth-degree Taylor polynomial

. Give the value of:

(a) (b) (c)

11. Use power series to evaluate

12. Use a Taylor series of degree 5 for sin x about x = 0 to estimate

with an error less than 0.001. Justify your answer.

13. The function f has derivatives of all orders for all real numbers x. Assume

(a) Write the third-degree Taylor polynomial for f about x = 3, and use it

to approximate .

(b) The fourth derivative of f satisfies the inequality for all x in the

closed interval [2.6, 3]. Use the Lagrange error bound on the approximation

to found in part (a) to explain whether or not can equal

(c) Write the fourth-degree Taylor polynomial, , for

about x = 0.(d) Use your answer to (c) to determine whether g has a relative maximum, a relative minimum, or neither at x = 0. Justify your answer.

14. The Taylor series about x = 4 for a certain function f converges to for

all x in the interval of convergence. The nth derivative of f at x = 4 is given by

(a) Write the third-degree Taylor polynomial for f about x = 4.(b) Find the radius of convergence.

(c) Use the series found in (a) to approximate with an error less than 0.02.

Review

15. Solve the equation when y(0)= 1. Ans:

16. Evaluate: Ans:

17. Find the slope of the tangent to the graph of x ln y + ex = y Ans: 1

18. The 4th-degree Taylor polynomial for , centered at x = 0 Ans:

19. Ans: diverges

Determine if the following series converge or diverge

1. Ans: D

2. Ans: D

3. , where k is a constant Ans: C

4. Ans: C

5. Ans: C

Answers to Review Sheet 3 on Series 1. Radius: ; Interval: 2. Radius: 1; Interval:

3. (a)

(b)

(c) or 0.321. Since the terms of the series are alternating in sign, decreasing in magnitude,

and having a limit of 0, the error is less than or equal to the third term, , which is less

than 0.001.

4.

5.

6.

7. (a) 0 (b) 30 (c) 480

8. a < 0, b > 0, c > 0 14. (a)

9. (b) 3 (c) 1.870. Since f is a convergent alternating

10. series with terms that are decreasing in

11. 2 magnitude and having a limit of 0,

12. 0.946. Since the terms of the series are alternating in sign, decreasing in magnitude,

and having a limit of 0, the error is less than or equal to the fourth term, , which is

less than 0.001.

13. (a)

(b) 0.005. Since – 6 does not lie in this interval, cannot

equal – 6.

(c)

(d) so g has a relative minimum at x = 0.