Chapter 7: Q. 80 (page 605)

Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.
If $r = 1$ then $r^{k} \to 1$

Short Answer

Expert verified

Hence proved, $r = 1 t h e n r^{k} \to 1$

Step by step solution

Step 1. Given information

The given sequence If $r = 1 t h e n r^{k} \to 1$

Step 2. To prove that the result,observe the behavior of geometric sequence forr=1

The geometric sequence $\{r^{k}\}$ with ratio $r = 1$ is a constant sequence with each term equal to 1

The term of the sequence $\{r^{k}\}$ is

$\{r^{k}\} = \{1, 1, 1 . . . . . . .\}$

The sequence $\{r^{k}\}$ is a constant sequence and is bounded

The constant sequence is always convergent and the sequence $\{r^{k}\}$ is converging to 1

Therefore, $r = 1$ then $r^{k} \to 1$ holds

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Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.
If $r = 1$ then $r^{k} \to 1$

Short Answer

Step by step solution

Step 1. Given information

Step 2. To prove that the result,observe the behavior of geometric sequence forr=1

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Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.If r=1thenrk→1

Short Answer

Step by step solution

Step 1. Given information

Step 2. To prove that the result,observe the behavior of geometric sequence forr=1

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Mechanics Maths

Statistics

Calculus

Logic and Functions

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Prove the statements about the convergence or divergence of sequences in Exercises 78–83, referring to theorems in the section as necessary. For each of these statements, assume that r is a real number and p is a positive real number.
If $r = 1$ then $r^{k} \to 1$