CALCULUS BC
REVIEW 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
6. Taylor series for
7. Suppose is approximated near x = 0 by a fifth-degree Taylor polynomial
. Give the value of:
(a) (b) (c)
8. Suppose is the second-degree Taylor polynomial
for the function f about x = 0. What can you say about the signs of
a, b, and c if f has the graph pictured on the right? Explain how
you got each sign.
TURN->
By recognizing the following as a Taylor series evaluated at a particular value of x, find the
sum of each of the following convergent series.
9. 10.
______
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.
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.