Module 21 - Sequences and Series

Module 21 - Answers

Lesson 1

Answer 1

21.1.1 The first ten terms of the sequence are

As n gets larger subsequent terms in the sequence get closer to zero.

Answer 2

21.1.2

The limit of the sequence as n approaches is 0.

Answer 3

21.1.3 The points in the graph level off and approach a horizontal line. In other words, the graph has a horizontal asymptote.

Answer 4

21.1.4 Plotting the terms of the sequence gives graphical evidence that the sequence converges. The window shown is [xmin, xmax] x [ymin, ymax] = [0, 20] x [2, 3] with nmin = 1 and nmax = 20.

A table of values of the terms of the sequence gives numerical evidence that the sequence converges.

The value of gives analytical proof that the sequence converges.

The sequence converges to e 2.71828.

Lesson 2

Answer 1

21.2.1

Answer 2

21.2.2 The fact that the graph appears to level off and have a horizontal asymptote suggests that the sequence of partial sums may converge, which would mean that the infinite series converges.

Answer 3

21.2.3

Because the limit of the ratio of the sequence of partial sums is less than 1, the series converges.

Answer 4

21.2.4 In the previous table the 26th and 51st partial sums were both approximately 10.5. So a reasonable estimate for the sum of the infinite series is 10.5.

Lesson 3

Answer 1

21.3.1