Chapter 7: Q. 22 (page 604)

Discuss the boundedness and monotonicity of the geometric sequence ${\{c r^{k}\}}_{k = 0}^{\infty}$ with $c > 0$ and $r < 0$ . In addition, determine whether the sequence converges or diverges. If it converges, find the limit of the sequence.

Short Answer

Expert verified

The sequence is not a monotonic sequence, unbounded and divergent.

Step by step solution

Step 1. Given information

We have to tell about the boundedness and monotonicity of the sequence ${\{c r^{k}\}}_{k = 0}^{\infty}$ with $c > 0$ and $r < 0$ .

Step 2. Determine the monotonicity and boundedness also determine whether the sequence converges or diverges.

$a_{k} = c r^{k}$

The ratio $r$ is negative because it is given $r < 0$ .

As the index $k$ increases, the sign of the sequence varies.

Thus, the sequence is not a monotonic sequence.

The given sequence has neither upper bound nor lower bound.

Therefore, the sequence is unbounded.

The sequence is neither monotonic nor bounded and hence cannot be convergent.

Therefore, the sequence is divergent.

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Discuss the boundedness and monotonicity of the geometric sequence ${\{c r^{k}\}}_{k = 0}^{\infty}$ with $c > 0$ and $r < 0$ . In addition, determine whether the sequence converges or diverges. If it converges, find the limit of the sequence.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the monotonicity and boundedness also determine whether the sequence converges or diverges.

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Discuss the boundedness and monotonicity of the geometric sequence crkk=0∞with c>0and r<0. In addition, determine whether the sequence converges or diverges. If it converges, find the limit of the sequence.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Determine the monotonicity and boundedness also determine whether the sequence converges or diverges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Theoretical and Mathematical Physics

Statistics

Calculus

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Study anywhere. Anytime. Across all devices.

Discuss the boundedness and monotonicity of the geometric sequence ${\{c r^{k}\}}_{k = 0}^{\infty}$ with $c > 0$ and $r < 0$ . In addition, determine whether the sequence converges or diverges. If it converges, find the limit of the sequence.