Question

A sequence $a_1,a_2,a_3.....$ has the limit $L$if the nth term $a_n$of the sequence can be made arbitrarily close to _____ by taking $n$ to be sufficiently _____. If the limit exists, we say that the sequence _________; otherwise the sequence __________.

Short answer

The value of limit of the given function is $L,\text{large},\text{converges},\text{diverges.}$

Step-by-step solution

Step 1. Given information.

Here, the sequence is given:

$a_1,a_2,a_3.......$

Step 2. Concept  used.

Here we use the concept of a sequence that diverges.

Step 3. Simplification.

Determine whether the sequence $a_n={(-1)}^n$is convergent or divergent.

If we write out the terms of the sequence, we obtain

$-1,1,-1,1,-1,1,-1......$

The terms oscillate between $1,-1$ infinitely often, $a_n$does not approach any number. Thus

$\underset{n\to \infty }{\lim }{(-1)}^n$does not exist; that is the sequence $a_n={(-1)}^n$ is divergent.

Therefore, our final answer in given gaps are :

$L,\text{large},\text{converges},\text{diverges.}$