Q. 81 - Calculus

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 $\left|r\right|>1,$then the sequence $\left\{r^k\right\}$ diverges and $\left|r^k\right|\to \infty$

Step-by-Step Solution

Answer

Hence proved,$\left|r\right|>1 then r^k\to \infty$

Step 1. Given information

The given sequence

If $\left|r\right|<1$ then the sequence $\left\{r^k\right\}$ diverges and ${\left|r\right|}^k\to \infty$

Step 2. To prove that the result,use the result of converges of geometric sequence with ratiio less than 1

Since,$\left|r\right|>1$

Therefore,

$\frac{1}{r}<1,$

The geometric sequence $\left\{\frac{1}{r^k}\right\}$ with the ration $\frac{1}{r}<1$ is convergent and converges to 0

Also,if $\underset{k\to \infty }{\lim }{a}_k=\infty ,$then $\frac{1}{a_k}\to 0$

The sequence $\left\{\frac{1}{r^k}\right\}$ is converges to 0.Therefore,

$\underset{k\to \infty }{\lim }\frac{1}{r^k}=0$

Therefore,

$\underset{k\to \infty }{\lim }{r}^k=\infty for \left|r\right|>1$(Because if $\underset{k\to \infty }{\lim }{a}_k=\infty ,then\frac{1}{a_k}\to 0)$

Therefore,if  $\left|r\right|>1 then r^k\to \infty$holds