Does the Sequence Converge or Diverge?

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In summary, sequence convergence refers to the behavior of a sequence as its terms approach a specific number or limit, while divergence refers to the behavior of its terms approaching infinity or oscillating without approaching a specific value. To determine if a sequence converges or diverges, methods such as the limit comparison, ratio or root tests, and direct comparison can be used. The convergence or divergence of a sequence is significant in understanding mathematical functions and series, as well as in various areas of science and engineering. A sequence cannot converge and diverge at the same time, and it is possible for a sequence to be neither convergent nor divergent if its terms do not follow a consistent pattern.
  • #1
karush
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$\tiny{s4.11.1.26} \\$
$\text{ Determine whether the sequence converges or diverges. If it converges, find the limit.} \\$
$$\displaystyle a_n=\frac{(-1)^n n^3}{n^3+2n^2+1}$$
$\text{ divide every term by $n^3$}$
$$\displaystyle a_n=\frac{(-1)^n }{1+\frac{2}{n}+\frac{1}{n^3}}$$
$\text{ take the limit}$
$$\displaystyle \lim_{{n}\to{\infty}} a_n=1$$
$\text{suggestions?}$
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  • #2
The limit of $a_n$ is not 1. If we replace $(-1)^n$ with 1, then indeed the limit is 1. But as it is, $a_n$ oscillate between numbers that are close to $-1$ and those that are close to $1$.
 
  • #3
so it diverges
 

FAQ: Does the Sequence Converge or Diverge?

1) What is the difference between sequence convergence and divergence?

Sequence convergence refers to the behavior of a sequence as its terms approach a specific number or limit. A convergent sequence will have all of its terms become arbitrarily close to the limit, while a divergent sequence will have its terms either approach infinity or oscillate without approaching a specific value.

2) How can you determine if a sequence converges or diverges?

One method is to use the limit comparison test, which compares the given sequence to a known convergent or divergent sequence. Another method is to use the ratio or root test, which look at the growth rate of the sequence's terms. Additionally, one can also use the direct comparison test, which compares the given sequence to a known convergent sequence with similar terms.

3) What is the significance of a sequence converging or diverging?

The convergence or divergence of a sequence is important in understanding the behavior and properties of various mathematical functions and series. It is also used in various areas of science and engineering, such as in modeling real-world phenomena and analyzing data.

4) Can a sequence converge and diverge at the same time?

No, a sequence cannot converge and diverge at the same time. A sequence can either converge to a specific limit or diverge to infinity or oscillate without approaching a specific value. It cannot exhibit both behaviors simultaneously.

5) Is it possible for a sequence to be neither convergent nor divergent?

Yes, it is possible for a sequence to be neither convergent nor divergent. This can occur when the terms of the sequence do not follow a consistent pattern or do not approach a specific value or infinity. In this case, the behavior of the sequence is considered to be undefined or indeterminate.

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