Question

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

Hence proved,$r=1then{r}^k\to 1$

Step-by-step solution

Step 1. Given information

The given sequence If$r=1then{r}^k\to 1$

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

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

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

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

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

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

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