Chapter 9.1, Problem 37E

To determine

The limit of a sequence if it converges.

Answer to Problem 37E

The sequence $a_n$ diverges and $\underset{x\to \infty }{\lim }{a}_n=\infty$

Explanation of Solution

Given Information: The sequence is defined as $a_n={\left(1.1\right)}^n$

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 $1.1$

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

  $\Rightarrow a_n\ge 1.1$ for all $n\ge 1$ (1)

Now, if for a large $N$ , $a_N={\left(1.1\right)}^N=\text{c}$ , then

  $a_{N+1}={\left(1.1\right)}^{N+1}=1.1c$

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

  $\Rightarrow c\approx 1.1c$

  $\Rightarrow c\approx 0$ [This contradicts the condition derived in (1)]

Therefore, the given sequence does not converge.