Science:Infinite Series Module/Units/Unit 1/1.1 Infinite Sequences/1.1.11 Example rn

Example

Determine the values of r so that the sequence

$\{r^n{\}}_{n=1}^{\mathrm{\infty }}$

is convergent.

Complete Solution

We can solve this problem by considering cases for the value of r.

Case 1

If r > 1, then  rⁿ tends to infinity as n tends to infinity. The sequence is divergent in this case.

Case 2

If r= 1, then

$\underset{n\to a}{\lim }{r}^n=1$

so the sequence is convergent for this case.

Case 3

If $-1<r<+1$, then

$\underset{n\to \mathrm{\infty }}{\lim }{r}^n=0,$

so the sequence is convergent for this case.

Case 4

If $r=-1$, then

$\underset{n\to \mathrm{\infty }}{\lim }(-1{)}^n$

does not exist, so the sequence is divergent for this case.

Case 5

If $r<-1$, then $r^n$ tends to negative infinity as $n$ does not tend to a single finite number. The sequence is divergent in this case.

Summary

Therefore, the sequence $\{r^n{\}}_{n=1}^{\mathrm{\infty }}$ is convergent when $-1<r\le +1$.

Explanation of Each Step

Case 1

Consider the case when $r=2$. Then our sequence becomes

$\{2,4,8,16,32,64,\dots \}$

which tends to infinity.

Case 2

Here we are using a fundamental property of limits, that the limit of a constant equals that constant:

$\underset{x\to \mathrm{\infty }}{\lim }c=c$

for any constant $c$ and $x\in \mathbb{ℝ}$.

Case 3

Consider the case when $r=1/2$. Then our sequence becomes

$\{1/2,1/4,1/8,1/16,1/32,\dots \}$

which tends to zero.

Similarly, if  $r=-1/2$. Then our sequence becomes

$\{-1/2,+1/4,-1/8,+1/16,-1/32,\dots \}$

which also tends to zero.

Case 4

In this case, we have the sequence

$-1,+1,-1,+1,\dots$

As $n$ approaches infinity the sequence does not approach a unique value, so the limit does not exist.

Case 5

This case is similar to Case 1. Consider the case when $r=-2$. Then our sequence becomes

$\{-2,+4,-8,+16,-32,+64,\dots \}$

The terms alternate between positive and negative numbers, and do not tend to a single finite number.

Possible Challenge Areas

Connecting Results to Definition of Convergence

In each of the cases, we used a limit to determine whether the sequence is convergent. According to our definition of convergence of a sequence, as long as our respective limits exist, then the sequence converges.