Q. 21 - Calculus

Question

Discuss the geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with $c>0$ and $r>0$ respect to its boundedness and monotonicity. Find the r values at which the sequence converges and the r values at which it diverges. Find the limit of the sequence if it converges.

Step-by-Step Solution

Answer

The sequence $\left\{a_k\right\}={\left\{c{r}^k\right\}}_{k=0}^{\infty }$ is convergent and the sequence if $\left|r\right|>1$then ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ is divergent.

Step 1: Given information

The geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$with $c>0$ and $r>0.$

Step 2: Calculation

The purpose is to discuss the monotonicity and boundedness of the sequence. ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$with $c>0$ and $r>0.$

The sequence $\left\{a_k\right\}={\left\{c{r}^k\right\}}_{k=0}^{\infty }$ has the general term $a_k=c{r}^k.$

The geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with ratio $r=1$ is a constant sequence with each term equal to c.

The terms of the sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ is ${\left\{c{r}^k\right\}}_{k=0}^{\infty }=\{c,c,c\dots \}$.

The sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ is a constant sequence and is bounded.

The sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ is convergent to c, and the constant sequence is always convergent.

Thus, the sequence${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with $c>0$ is convergent for $r=1.$

The geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$with ratio $0<r<1 :$

It has been noted that

$-\left|c{r}^k\right|\le c{r}^k\le \left|c{r}^k\right|$

If $\left|r\right|<1,$ then

$a_{k+1}=\left|r\right|a_k$

     (or)

$0\le a_{k+1}<a_k$

The decreasing sequence  $\left\{ak\right\}=\left\{c{r}^k\right\}{k=0}^{\infty }$ is constrained below by 0.

Convergence occurs in the monotonically decreasing sequence that is bound below.

As a result, the sequence$\left\{a_k\right\}={\left\{c{r}^k\right\}}_{k=0}^{\infty }$  is convergent.

Step 3: Further Calculation

Assume that $a_k\to l$.

Therefore, $a_{k+1}=\left|r\right|a_k$

$\underset{k\to \infty }{\lim }\left(a_{k+1}\right)=\underset{k\to \infty }{\lim }\left(\left|r\right|a_k\right)$ (Take limit)

$l=\left|r\right| l$ (Because $a_k\to l$, then  $(\left.a_{k+1}\to l\right)$)

$l(1-|r\left|\right)=0$( Transpose )

$l=0$,$\left|r\right|=1$

$l=0$ (Because $\left|r\right|<1$)

As a result, for $\left|r\right|<1$ the sequence $\left\{a_k\right\}={\left\{c{r}^k\right\}}_{k=0}^{\infty }$ converges to $0$.

The geometric sequence ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$ with ratio$\left|r\right|>1$.

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

Therefore,

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

The geometric sequence $\left\{\frac{1}{r^k}\right\}$ with ratio$\frac{1}{r}<1$ 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 $\left\{\frac{1}{c{r}^k}\right\}$ is converging to 0, therefore,

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

Consequently$\underset{k\to \infty }{\lim }c{r}^k=\infty$ for $\left|r\right|>1$ 

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

As a result, if $\left|r\right|>1$ then ${\left\{c{r}^k\right\}}_{k=0}^{\infty }$is divergent.