Chapter 9.1, Problem 38E

To determine

The limit of a sequence if it converges.

Answer to Problem 38E

The sequence $a_n$ converges to $0$ .

Explanation of Solution

Given Information: The sequence is defined as $a_n={\left(0.9\right)}^n$

Concept Used: A sequence converges if and only if the limit is finite and single valued.

Each term of the sequence is equal to its preceding term multiplied by $0.9$

It means, each term of the sequence is less than the preceding term.

  $\Rightarrow a_n\le 0.9$ for all $n\ge 1$ (1)

Now, if for a large $N$ , $a_N={\left(0.9\right)}^N=\text{c}$ , then

  $a_{N+1}={\left(0.9\right)}^{N+1}=0.9c$

But, for the sequence to converge, when $N$ is sufficiently large ( $N\to \infty$ ), $a_N\approx a_{N+1}$ .

  $\Rightarrow c\approx 0.9c$

  $\Rightarrow c\approx 0$

Therefore, the given sequence converges to the limit $0$ .