Math 2224

Review - Sequences and Series

Sequences

{a_n} = a₁, a₂, a₃, a₄,...

A sequence converges if

ex. For the sequence where , the first four terms are: . The limit as n approaches infinity is 1, so the sequence converges.

Problems: Write the first four terms of each sequence. Determine whether each one converges or diverges. If it converges, find the limit.

1. 2. 3.

Series

Special series: geometric, power (Taylor, Maclaurin), telescoping, p-series, alternating series.

Tests for convergence: nth term test, altenating series test, direct comparison test, limit comparison test, ratio test.

The Maclaurin series for sin t, cos t, and e^t converge to their respective functions for all t. You should know the following:

1)

2)

3)

Also note that in fact the Taylor series at c (where c is any real number) for the functions sinx, cos x, and e^x converges to the functions for all x.

Problems:

1. Determine the convergence or divergence of each of the following series. State which test you are using. Is problem g absolutely convergent?

a) b) c) d)

e) f) g) h) i)

2. Find the sum of each of the following series: .

3. Approximate the sum of the following convergent alternating series with an error of magnitude ≤ 0.001.

4. Find a positive integer n such that approximates the sum of the series with an error of magnitude ≤ 0.0001.

5. Use the Integral Test to approximate to within 0.1.

6. Find the radius of convergence and interval of convergence of each of the following power series.

a) b) c)

7. Find the series interval of convergence and, within this interval, find the sum of the series as a function of x.

8. Find the first four terms of the Taylor series for f(x) = cos x at c = π/3.

9. Find the Maclaurin series for f(x) = ln(x+1) by finding a pattern for the successive derivatives at 0 to find an expression for the nth term of the series.

10. Find the Maclaurin series for following function and write out the first six nonzero terms of the series:

11. Use a series representation to approximate to four decimal places .

12. Use the first six terms of the Maclaurin series to approximate and estimate the error in the approximation.

13. Find the Maclaurin series for the function and write out the first six non-zero terms of the series.

14. Use a series to evaluate correct to 4 decimal places;

15. Find the first three terms of the Taylor series at c = π/3 for the function f(x) = sec x.

Answers

Sequences

1. diverges 2. converges, 3. diverges

Series

1. a. converges, limit comparison test b. diverges, limit comp. test

c. diverges, nth term test d. diverges, geometric series

e. converges, direct comp. test f. converges, ratio test, absolutely convergent

g. diverges, limit comp. test h converges, direct comp. test

i converges, ratio test

2. a. -1 b.

3. 0.902116

4. 99

5. 1.162

6. a. r = ∞, (-∞,∞) b. r = 1/5, [24/5,26/5) c. r = 0, series converges only for x = 3

7.

8.

9.

10.

11. .4969

12. .36666, error approximation a₇: 0.00138889

13.

14. .0090

15.